I'm a little confused by the reference to healthcare.gov: if Big Web is not the right paradigm to handle a... big web site, then what is?
I realize that’s an underwhelming answer, but if we asked that everyone who disagreed with the status quo simultaneously proposed a perfect solution to fix it, we would be nowhere in any sense. I assure you that thinking about the answer to your question is a significant part of my life, and talking to people who use computers to shape that thinking is key. Government was largely responsible for setting the computing agenda and spurring all of the right technologies in the beginning, and I believe there is merit to revisiting that thinking rather than ceding computing to engineers on the West coast.
There were precursors to this... GWT was an early example of abuse of the web browser stack to accomplish something very non-webby. But it's really gone full bore now.
Back in the mid-90s when I made web pages for a living when some designer came to us with a photoshop mockup in a PSD and said "make the page look like this" we felt perfectly fine saying "no, the web can't do that. design for the web, not photoshop"; no such luck now. And so here we are...
There is a watershed coming where Big Web/FAANG will lose this mantle, and DevOps, half-baked Agile because nobody ever bakes it all the way, framework engineering, and a complete fundamental ball-dropping around security and exploitation of user data are rapidly ensuring it.
At some point, Big Web/FAANG and everybody delivering Web services of some kind got the idea that the Web was computing. I think the prototypical version of this thinking led to Rob Pike's observations about systems research in the early oughts. This forum tends to operate with that underlying assumption about the lens to view computing through no fault of anyone in particular, which is why I say that.