Regarding the disconnect issue, while may be well and fine if one play on a well setup desktop, not every one play in the same condition and setup.
Many times, battery powered devices do disconnect due to low charge and seem increasingly popular. Also bad connections happens all of the time and with battery powered controllers like the PS4 and XBox, the micro-USB port is not exactly the most rugged connector out there, specially after a few years of use and/or a few drops.
I can't count the times the doggo got a bit too lively while playing with a toy and tripped on some cable, like the laptop charging cable.
Shocker650: The problem is that my controller can just simply lose connection for a moment, reconnect and I need to quit the game and start it up again in order for my controller to work again. This is not an issue on Steam, so why can't GOG fix it?
That may not be a issue related with GOG but with the game itself.
Plenty of games on GOG that you can unplug and reconnect the controller and still works fine. I'm just speculating here but some old games use a emulator/wrapper to work (DosBox or similar) and the "emulator" is the program that is translating the controller input to keyboard keys.
As asked before, care to share what games have this problem?
Edit: just checked your profile and the list of games you play are mostly old like Resident Evil 1/2, Swat, Mortal Kombat etc. As mentioned, those are old games and the wrapper needed to make them work on modern windows, might not be able to "plug-and-play on the fly", it might be GOG's fault since they were involved with Resident Evil re-release, but not sure about other games.