I think either direction would be interesting. Or both. The browser was an experiment in the first place, but people seem to have stopped experimenting and are just doing the equivalent of what building on an IBM mainframe would have been in those days. It's disappointing. (It's not unlike people who quote Martin Luther King today, not appreciating that the essence of his radicalism was the direction of travel towards justice, not the political compromises themselves which are now firmly established.)
I hope at least that the renewed interest in systems programming that's come with the Rust fandom might spur some actually innovative developments to replace the antiquated model we're all stuck using. But sadly the tech community seems to be split into two communities, one of which has no interest in innovating, and the other of which seems to be set on innovations so absurdly impractical (¡world wide web on the blockchain!) that they could never take off.
Alternatively, the modern browser has basically evolved into a virtual machine anyway. You seem to be suggesting that we could do exciting things with a more intentional version of that idea, and I don't disagree.