I can't think of how would an OS be in charge of what the browser does

Yeah, me neither and is what makes it so strange. Brave does alert me to update it every time I open it up, but the update isn't automatic nor does it work in the software manager so I have just kind of been leaving it alone since it works like 95% of the time.

I installed all the software either from the software manager or if I couldn't get that to work by copy/pasta in the terminal. I honestly don't remember which but it wasn't flatpaks

It's not a big deal though, almost all the time it works fine just some days it decides it doesn't want to play. I suppose it is still leaps and bounds better than my WinOS that had some sort of issue on almost a daily basis.