I was watching a video of a player playing my game, and they clicked on one of the side buttons while playing the webGL version, and it ended their session. Maybe there's something I could do to protect against this but it would be nice if somehow it wasn't so easy for players to run into this. Maybe I can remove these from my page as well? 

I’ve noticed that too, there are games that have been part on a variety of events (especially gamejams with long names) and the buttons on the top-right eat a lot of space.

I believe the solution for developers is to just use a banner. Not elegant, but should work in most cases. So the "side buttons" are over the banner and then the game area begins.

A sort of solution on desktop is to just tell your browser to display that area different.

This should work as a user css style.

@-moz-document regexp("^https://[^/]+\\.itch\\.io/.*") {
.user_tools {
    position: static;
}
}

What also should work, is zooming in, till the button area gets flattend on top. The Itch css is adaptive to screen width. But that does not work on some games, as they need default or even lower zoom.