I've been having this issue too--a new battery didn't help me.
I'm convinced it is software based. After I finally got fed up with the issue I rooted and installed a CM 13 rom and everything was fine, no random restarts anymore (i.e. I no longer think it was battery), but I couldn't use NFC while rooted and I use it a lot, so I went back to unrooted stock rom, and the random restarts happened again.
After I went back to stock, I tried buying the new battery. My battery life has been a bit better on the new one, but the random restarts still happen. The other strange thing is that after the restart, it doesn't clear the cache (you can open the recent apps tab and everything is still in there), but an actual restart would--so it seems tied to how the software is managing memory or power somewhere.
None of this happened on previous versions of the stock rom either, so it seems this is a 5.0 issue for the G3.
Edit: although, a caveat to my issue--this happens while the phone is in use (random reboot while I am browsing, email, etc.)--maybe my issue is a bit different since it sounds some here don't even get past boot screen.