You'd never design from the start the bonkers HTML/CSS, box model, and other junk we are stuck with, but browsers have a number of good aspects that you'd probably end up with in a complete redesign.
The problem is multifaceted. UI toolkits are very hard to do well (I have a lot of opinions on this, and my own crossplatform toolkit), even on a single platform. Just that alone is huge. There are basically no overall great UI toolkits, even if some have good parts. Even a theoretical great UI toolkit will have quite a task to achieve parity with what browser UIs can do.
If we could get past that (spoiler: we can't, no way in hell) then we'd also have to be willing to ditch all the legacy browser stuff to switch billions of users and devices to something new AND we'd still have many other big problems to deal with: getting the committee/everyone to agree without ruining it, security, privacy, etc.
We've accepted we can't really fix it, so the only thing we can do is keep extending it. That's how we got here.