I finally figured out the problem. The clue was when I uninstalled Opera and installed Chrome only for the random closures to keep happening. That's when I installed Avast and Malwarebytes and did a virus scan and sure enough, it found potential keyloggers and a few other suspicious programs. After deleting the viruses Opera now runs fine with all my extensions and I haven't had a problem since.
I have no idea why these viruses were constantly shutting down Chrome-based browsers but they're definitely the culprit. If you're reading this message from the future experiencing the same problem now you know--don't just rely on antivirus tools that come with Windows and try running a third party program!