Thanks for your replies, guys.
@Lando242, I did a clean install, but that didn't work. I checked the option ''Delete my Opera user data'' in the uninstaller and went to the profile directory to make sure the folder was deleted (and yes, it was).
@L33t4opera, I already have that flag enabled. It was working before the update, now it has no effect.
My guess is that something is preventing the GPU process to be launched on Opera's startup. As I said: only Opera 28 and 29 are affected (for now), so I assume that it's some kind of unidentified bug inside the browser's code. That would explain why the flag doesn't work and why the GPU process is not listed under the task manager.