The choice isn't between "support HTML/CSS/JS" and "don't support HTML/CSS/JS". No Web browser can drop backwards compatibility. We haven't even been able to get rid of backwards compatibility for much worse things than CSS, such as <center>, because sites you need to use (such as Hacker News) won't update their markup to use CSS for layout.
asm.js is attractive because it is a small extension to the Web platform, so the overall additional complexity of an HTML/CSS/JS engine (which we are likely never going to be able to drop) doesn't go up much by implementing asm.js optimizations.
We're talking about Web browsers. Technologies that try to integrate into Web browsers specifically but don't have a good compatibility story with the platform haven't had a lot of success.
In some ways they do, but they are also nonstandard, with the downsides that that brings. Neither overall approach is perfect, it's good that we have a combination of both in our field.
Why get hung up on the fact that something is non-standard? Things don't just pop into existence as a standard way of doing things. Some core group of people has to agree on them and then away we go - we have a standard!
That's why everybody should actively use non-standard things that they want to become standards.
asm.js is attractive because it is a small extension to the Web platform, so the overall additional complexity of an HTML/CSS/JS engine (which we are likely never going to be able to drop) doesn't go up much by implementing asm.js optimizations.