it only manages one stream at a time
I'll take flak for saying it, but I feel web developers are partially at fault for laziness on this one. I've often seen them trigger a swath of connections (e.g. for uncoordinated async events), when carefully managed multiplexing over one or a handful will do just fine.
Eg. In prehistoric times I wrote a JavaScript library that let you queue up several downloads over one stream, with control over prioritization and cancelability.
It was used in a GreaseMonkey script on a popular dating website, to fetch thumbnails and other details of all your matches in the background. Hovering over a match would bring up all their photos, and if some hadn't been retrieved yet they'd immediately move to the top of the queue. I intentionally wanted to limit the number of connections, to avoid oversaturating the server or the user's bandwidth. Idle time was used to prefetch all matches on the page (IIRC in a sensible order responsive to your scroll location). If you picked a large enough pagination, then stepped away to top up your coffee, by the time you got back you could browse through all of your recent matches instantly, without waiting for any server roundtrip lag.
It was pretty slick. I realize these days modern stacks give you multiplexing for free, but to put in context this was created in the era before even JQuery was well-known.
Funny story, I shared it with one of my matches and she found it super useful but was a bit surprised that, in a way, I was helping my competition. Turned out OK... we're still together nearly two decades later and now she generously jokes I invented Tinder before it was a thing.