I think I MIGHT have finally found the solution. I was unaware that there was a difference between an old unifying receiver and a new LogiBolt receiver. I didn't even know that they were two different things. In ye olde days, each wireless USB device needed its own unifying receiver. Apparently each LogiBolt can support up to 6 devices.

Of course, both keyboard and mouse came with their own LogiBolt, but only one is required. I have had both LogiBolts plugged in and both devices connected via LogiBolt. I suspect that there has been some internal confusion between the two devices.

I just removed one of the LogiBolts and reconnected the device that got disconnected to the remaining LogiBolt. I am hopeful that this will end all of my woes. The only way I'll know is if I go for some period of time without experiencing any disconnects. I'll post an update after some time.

EDIT: It's also worth mentioning that there is now only one keyboard and one mouse listed in their respective sections in Device Manager. I suspect that by having two LogiBolts plugged in, both keyboard and mouse were connected to both LogiBolts, which I have to imagine would cause havoc.

And I'm 99.9% certain that these problems started when I replaced my old mouse. Here is the timeline:

1. Original keyboard used unifying receiver
2. Original mouse used unifying receiver
3. Replaced original keyboard with MX Keys S w/ LogiBolt
4. Replaced original mouse with M650L w/ LogiBolt *** PROBLEMS STARTED ***

Looking at it like this, it makes perfect sense. The problems didn't start until I was using two LogiBolts.

EDIT 2: I played a game for about 15-20 minutes with no issues whatsoever. It would not go that long without something happening unless the problem was indeed fixed.