However, for better or for worse, it is not the Linux way.

Maybe it should be, but it isn't.

I can't think of a single Linux app that self-updates this way. I suspect that on most distros, permissions issues would prevent it.

I also suspect that if it were easy, Google would have done it. It is not a company that does things the hard way for fun.

Packaging apps as distro-native packages isn't trivial but there's tons of existing tooling for it. Ditto for hosting repos.

So what Google did is package stuff so that it adds its own repos when you install it, and then it is left up to the OS to keep it updated.

That to me says that that was an easier job than writing self-updating Linux apps, like Chrome on Windows or on macOS.