Many factors and capabilities went into Firefox's success. While it's easy to enumerate the primitives required in hindsight, I'm doubtful that if OSes of the period had taken a restrictive stance based on contemporary ideas of what should be allowed, that an easy time would have been had.
This is not hypothetical, consider the present. While Firefox on iOS exists, it's just a branding skin over WebKit, due to a similar flavor of security paternalism around JITing code (only bad people write self modifying code :-). If Firefox had needed to differentiate itself originally in such a market, it's doubtful it would have had much success.
A threading free-for-all may be the wrong abstraction to use for many applications, but it has the virtue of being a decent stand-in for the hardware's actual capabilities. It's also close enough to ground truth that most other abstractions can be built on top of it. Imagine how unpleasant building a browser on top of Workers + ArrayBuffer transfer would be (especially given the lousy multi-millisecond latency of postMessage in most browsers). Also, consider that while there is often loud agreement that raw threads are dangerous, after decades of research, there's little consensus on the "right" option amongst many alternatives.
SharedArrayBuffer is nearly as powerful as the proposal, but not quite.
For example, while it allows writing a multi-threaded tree processing library, it would have trouble exposing an idiomatic JS API if the trees in the library live in a single SAB (as JS has no finalizers etc. to enable release of sub-allocations of the SAB). The options are either one SAB per tree (which likely performs badly), an API where JS users need to explicitly release trees when done with them, or leaking memory. With the proposal, each tree node could be represented directly as a JS object. The proposal may not be the best way to fix this problem, but we definitely still have gaps in JS concurrency.
Agreed this would be a serious undertaking, however, and not to be lightly considered.
The proposal goes a long way to make the case this can be implemented performantly, but some deep thought should go into how it would alter / constraint future optimizations in JS JITs.