I think it's a useful thought experiment. Some reasons why it seems like a bad idea:
a) Bugs. Browsers are already pretty much the most complex software out there, and they have tons of bugs. A meta-browser would be orders of magnitude more complex, and so likely have orders of magnitude more bugs.
b) Security. Special case of a) above. Browsers have had decades to gradually iron out security issues, and we still see periodic flare-ups. A meta-browser would start from scratch and be taking on a much bigger problem. You would need some way to allow people to add new engines without causing them to insert malicious code, interfere with other websites' browser engines, steal people's information, etc. It's hard enough protecting people when all malicious websites have is html and javascript. It's much worse when they have the ability to write Assembly. See https://en.wikipedia.org/wiki/ActiveX for an earlier attempt at this.
c) Adoption. Writing a browser engine is a lot of work, which is why there hasn't been a new one in a decade. If every web app had to provide a browser engine nobody would build web apps for your meta browser. Everyone would end up just using one of your default engines, at which point you're back to the current state of the world but with all the complexity of points a) and b) above.
It's worth backing up and asking yourself: what is the problem you're trying to solve. Then we can talk about whether it's really a problem and what the solution might look like, without immediately barreling down the first solution that comes to mind.
a) Bugs. Browsers are already pretty much the most complex software out there, and they have tons of bugs. A meta-browser would be orders of magnitude more complex, and so likely have orders of magnitude more bugs.
b) Security. Special case of a) above. Browsers have had decades to gradually iron out security issues, and we still see periodic flare-ups. A meta-browser would start from scratch and be taking on a much bigger problem. You would need some way to allow people to add new engines without causing them to insert malicious code, interfere with other websites' browser engines, steal people's information, etc. It's hard enough protecting people when all malicious websites have is html and javascript. It's much worse when they have the ability to write Assembly. See https://en.wikipedia.org/wiki/ActiveX for an earlier attempt at this.
c) Adoption. Writing a browser engine is a lot of work, which is why there hasn't been a new one in a decade. If every web app had to provide a browser engine nobody would build web apps for your meta browser. Everyone would end up just using one of your default engines, at which point you're back to the current state of the world but with all the complexity of points a) and b) above.
It's worth backing up and asking yourself: what is the problem you're trying to solve. Then we can talk about whether it's really a problem and what the solution might look like, without immediately barreling down the first solution that comes to mind.