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Question: can USB devices (keyboard, mouse) that don't seem to shutdown (lights are on) along with the system kill PSU?

Too short; I want read more:

Once upon a time I had Thermaltake Toughpower 750W PSU. It worked well for many years, but at some point it started to shutdown quite randomly. I couldn't track the problem for a long time, because it wasn't really consistent with power load I put my system under, though it clearly was correlating with weather/room temperature.

Eventually I knew something was actually overheating, but that "something" wasn't included into the PSU's thermal control and heatsink circuits. So I guess the developers didn't think it would overheat at all?

The thing is it happened in a couple of years (I think) after I bought Razer's keyboard (backlighted) and mouse (backlighted logo). The keyboard often keeps its lights on when I shut down the PC, and the mouse is always lit.

Once I thought it's enough, I bought a new 750W PSU. Okay, problem solved. But after a year or two I noticed a strange behavior (which I think initially wasn't there). The new PSU spins its fan like crazy reacting on high CPU load (i5 4670K 84W TDP), but barely reacts if GPU (Fury X 250W TDP) is running for hours (like, mining). So, again it's inconsistent with the actual power load. Dust is not the issue, PSU isn't blocked or anything.

As far as I know there is some low power 5V circuit for USB devices that are powered in stand-by mode. Is it possible the devices draw too much power and ruin some circuits while PC is shut down and PSU can't cool itself properly?

It is highly unlikely a USB device is the source of your issue. Motherboards monitor the power draw of the different power rails and will shutdown if they detect if the voltage is significantly out of normal operating range. In addition to that, USB controllers will shut down a port, or the entire hub if they detect a device is drawing to much power.

Power related issues can be hard to diagnose. This is even more true if you aren't getting clean power. Try moving your computer to a different power outlet, or even elsewhere in your home. It might not be a bad idea to invest in a UPS to ensure your computer is getting consistent and clean power.

Highly unlikely.

USB devices typically use only a few watts of power, hard drives generally being the biggest users. A keyboard and mouse would use much less. Much less. For a 750 watt PSU this is nothing.

I have a PC that runs 24/7 and has done so for more than 6 years. That is fully powered up and running. The PC wasn't new when I bought it and The PSU isn't 750 watts. Any decent PSU can do this and millions do.