I can see a very good reason to have it there:
It creates a unique thing that no other app has, to help you remember to close OBS with THAT button and not anything else.
If you use the "X" button in the corner of the window like everything else has, then you can easily end up playing "whack-a-mole" with multiple windows, and then the setting to remember them for the next session doesn't work.
And the File -> Close or File -> Exit method seems quaint to me, and probably a lot of other people too. Keep it there, because it feels incomplete without it, and of course it should work, but it's barely ever actually used or even thought of.
As for hitting it by accident, the closest fat-fingerable thing is Settings. Are you really going there that often? In a panic so that you're not paying close enough attention?
If there's something else next to it that you're fat-fingering instead, keep in mind that you can move these things around. See mine, for example:
Kinda hard to fat-finger that, except for Settings, and I do enough testing beforehand that I really don't need Settings when I'm live. Likewise for the entire stack, really, except for the Streaming and Recording buttons, but they're WAY up there!
