Wouldn't that make them entirely beholden to search engines that rely on ad revenue to pay Chrome? It doesn't seem to change the underlying incentives very much.

I'd say there's a big difference between having one primary customer and literally being owned by that customer. I would bet that a significant part Chrome's development is built around feeding data to other parts of Google, which would be more difficult and less appealing as a separate company.

Plus, there's no rule that says a browser has to get most of its revenue from selling Google Search placement. That just happens to be the case for the single independent browser that exists right now, one which isn't exactly renowned for great business leadership. I think if the Chrome team had a reason to explore self sustaining revenue models there's a good chance they would come up with some solid innovations.

Doesn’t that already happen? Chrome currently survives thanks to advertising either way. Splitting Google would just make it a little more indirect.

It currently is completelly owned by Google (Alphabet) so your comment would be more of a "there is a possibility that they would end up still relying on Google's ad revenue".