One natural place is for packaging apps. I have a lot of modern web apps packaged so they run as if they were separate applications rather than my main browser. A browser that was focussed on doing this well would still be very useful even if it only worked with the most up to date sites.

Lots of game UIs use internal browsers, which again could be another niche.

And, especially for languages with limited UI capability, a good embeddable browser could provide a decent way to build UIs. Again a niche where backwards compatibility is not so important.

I'd be pretty happy to make a split where I use one browser for document consumption and a totally different one for applications, and perhaps yet another for really old school sites. In an ideal world they could launch each other for the appropriate sites.