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Einstein's mathematical model of brownian motion furnished strong support of the atomic model but did not furnish airtight proof of its uniqueness (that is, the nonexistence of alternative models) at the time it was proposed.

It is worthwhile to note that it wasn't his objective to logically exclude the possibility of alternative models but rather to demonstrate that the atomic model furnished a quantitatively consistent explanation of the phenomenon.

His work was an important piece of a larger puzzle, contributed to by a variety of other researchers in the field, and once his piece of it was in place, the rest of the picture came into better focus- and it became much more difficult to successfully argue against the atomic hypothesis.

A quote from a physics textbook, which I read years ago, has always stuck in my head. It said:

"Some of the material in this book is undoubtedly incorrect. But if we are really lucky, MOST of it is incorrect".

This is because, in one sense, you raise a perfectly valid point in your question.

There IS a theory that describes Brownian motion more accurately than our current model of atoms jostling a given particle.

This theory would improve on atomic theory by telling us, for example, why the elementary particle that comprise the atoms have the masses, electric charges and other physical properties that currently we can only experimentally measure.

This theory would also explain other phenomena and physical constants that we don't currently have a satisfactory explanation for, It would also be able to reconcile gravity with the standard model of particle physics , which we cannot yet do.

As you may know, our current standard model can only deal with around four percent of the mass-energy in the universe, with the rest consisting of dark matter and dark energy that are both invisible and almost impossible to detect.

A theory that explained the precise process behind the origin of the universe, and it's ultimate fate, would also be very nice to have.

Unfortunately, as you know, this theory has not yet been developed....

But it is the role of physics to work continually towards such a theory. As part of this process, our current atomic model, which is very accurate in describing a limited set of physical phenomena and behaviour, would be incorporated into this newer model of reality.

An example of this process, developed over the last century, is General Relativity, which explains the universe much more accurately than our previous theory, (the Newtonian Model). But we still use Newtonian physics to guide space. probes to land on Mars, in the same way that we can use our current atomic theory to explain Brownian motion.

Personally, I don't ever expect a full / complete theory linking all aspects of physics. We have more questions than ever today. My intention in this answer is to argue that an "asymptotic" (sorry, I can't think of a more appropriate word) process, achieving better and better models, each new one based on the previous model, is hopefully what will occur in the (far) future.

I think that the best answer to this question was given by Einstein in the introduction of his 1905 paper on the theory of brownian motion (the title was actually "On the movement of small particles suspended in a stationary liquid demanded by the molecular kinetic theory of heat", which is already quite illuminating):

If the movement discussed here can actually be observed (together with
the laws relating to it that one would expect to find), then classical
thermodynamics can no longer be looked upon as applicable with
precision to bodies even of dimensions distinguishable in a microscope
: an exact determination of actual atomic dimensions is then possible.
On the other hand, had the prediction of this movement proved to be
incorrect, a weighty argument would be provided against the
molecular-kinetic conception of heat.

(from the 1956 Dover edition of the paper, translated by A.D. Cowper).

It was crystalline clear to Einstein (who did not know too much about brownian motion at the time he wrote the paper) that the validity of his theory would not had been a direct proof, but its falsification would had been a very strong argument against the atomic theory. Theory that, at the beginning of the twentieth century, was still considered by many mainstream physicists as a tool for computing chemical equilibria but lacking any other experimental evidence.

Nothing in any science can be proved absolutely.

Atomic theory is our current best guess to explain brownian motion, among other things (such as radioactivity, all of chemistry, etc). In that sense, brownian motion supports atomic theory, but does not absolutely prove it.

One day, as StudyStudyStudy suggests, we will have a better theory, which is less wrong about the universe. However, we'll likely still use atomic theory to explain brownian motion, because it works. No need to go into quantum theory to investigate a game of pool.

That's the loop of science: theory, falsification, better theory. Because of this, we'll never reach a complete theory, and if we do, we won't know it. That's what keeps physics interesting!

EDIT: bonus philosophical argument!

How can change exist? If something changes, it's not that thing anymore, so it didn't change, so change is impossible. Early Greek philosophers saw two ways out:

a) We exist in a "block universe" where everything's one big brick and nothing happens. That's clearly not the case, so

b) The universe is made up of many tiny unchangeable bits, which make up different objects when put together in different ways — atoms!

This is just one example of falsification at work, even in the early transition from philosophy to science.

Brownian motion itself did not prove anything about atoms until Einstein in 1905 formulated a theory that connected the two. Einstein did the following: He wrote down a molecular model for Brownian motion, obtained the relationship between the diffusion coefficient (a measurable quantity) and molecular size (unknown at the time) and used the theory to calculate the size of a molecule from its diffusion coefficient. This does not prove uniquely that atoms exist. It proves that atomic theory under reasonable assumptions can explain Brownian motion.

Brownian motion is intimately connected to the existence of atoms in that the diffusion coefficient and its theory, formulated by Einstein, rely on it to make quantitative predictions. If you mix up the idea that atoms exist and we can study them from a statistical point of view and that their effect is measurable also on a very small scale (micron) you get diffusion. Boltzmann devised the theory. Einstein had the wonderful idea of assuming that the theory was not only applicable to atoms but also to micron-sized objects, which are big enough to be observed but small enough to be influenced significantly by atoms. There is a specific step, easier to see in Langevin's idea of writing an explicit stochastic equation for Brownian Motion, in which Einstein assumes that micron-sized particles are in thermal equilibrium with the sorrounding atoms, i.e. they follow the equipartition theorem so that their squared velocity is $K_B T / M$ (where $K_B$ is Boltzmann's constant, $T$ the temperature and $M$ the mass of the particle. Notice that this like saying,scaling up, that mosquitos hitting a hot air balloon have an effect on it so that everything is at equilibrium and that this effect could be macroscopically measured (actually, if you had a mole of mosquitos, this would not feel so crazy, right..?).

Anyways the point is that the theory of BM shows that the statistical behavior of atoms can be observed by looking at bigger objects. Leaving philosophical interpretation aside, brownian motion allows for an independent measurement of avogadro's number. The only ingredients are
1)atoms exist
2)micron sized objects obey statistical mechanics

Such a measurement was (very smartly) done by Perrin (nobel price). Measuring the sedimentation of micron-sized particles he was able to estimate the diffusion coefficient $D=mu R/N_A T$ (where we see the mobility, the gas constant, temperature and, as the only unknown, Avogadro's number).
His result was very close to already estimated values of $N_A$ thus proving that the theoretical framework of physics based on the atomistic hypothesis was "true" or at least experimentally testable.

Notice that most of the merit goes to Einstein and Boltzmann for the theory of diffusion and statistical mechanics. But it was actually Perrin doing the measurement and proving everything as the theoretical part per se was very controversial at the time.

To explain this from a more mathematical standpoint one has to understand what prove means. I mean we know what it means so I won't go into that, but what one has to understand is that even in Mathematics nothing is proven from a vacuum. There is always some assumption. For instance here is one assumption given below:

The universe is deterministic and therefore any two universes with the exact same configuration will repeat the same behavior.

Now obviously the existence of other universe is not implied or denied here. The claim here is merely that if the copied at the exact moment an apple fell out of a tree towards Newton's head that both in the original and in the copy the apple will hit Newton in the head and he will remember to include gravity in his calculations.

What if that were one day proven to be false? There may be some things that are deterministic but there might be some 1% of the universe that isn't just random in the pseudo-random "I have a 50% of flipping a coin and getting heads" sense but truly 100% random in how particles interact. Is that likely? It is probably not the case, but it's an extreme counterexample.

The other issue is that science is done with that assumption of course but it also takes discrete measurements during experiments and attempts to construct models based on that. Even if we were to consider something like the path something follows when rolling down a hill in some repeated experiment we will always have a finite number of data points but the actual curve and representation is continuous (another assumption but safer). But now consider that with anything in science really. We might talk about large objects or small objects but the issue is the same in that we only observe the large picture not the fine details.

So when you ask if brownian motion proves the existence of atoms, no it does not. However, observing an atom under a microscope and naming that object as "an atom" does likely serve as proof. It doesn't prove the properties of that object. It merely proves that there is that thing that we call an atom. In that same sense I can prove water exists or air exists. But no Brownian motion could only prove atoms if there was truly no other option and the data points involved somehow happened to narrow down the theories to that one case.

There are infinite theories to explain any finite amount, and therefore selecting any single theory for the behavior of the universe requires infinite data.