Not really? Sort of? I don't really have a good answer here. It depends on what you mean by "due to". It's certainly due to the impulse response, since in the sense I meant "far worse" the impulse response is the only thing that matters.
Truncating the impulse response after five time constants wouldn't really change its output noticeably, and even if you truncated it after two or three time constants it would still be inferior to the box filter for this application, though less bad. So in that sense the problem isn't that it's infinite.
Likewise, you could certainly design a direct-form IIR filter that did a perfectly adequate job of approximating a box filter for this sort of application, and that might actually be a reasonable thing to do if you wanted to do something like this with a bunch of op-amps or microwave passives instead of code.
So the fact that the impulse response is infinite is neither necessary nor sufficient for the problem.
The problem with the simple single-pole filter is that by putting so much weight on very recent samples, you sort of throw away some information about samples that aren't quite so recent and become more vulnerable to false triggering from a single rapid mouse movement, so you have to set the threshold higher to compensate.
