What about a dystopian future where sites only work via SGX-like secure compute environments that remotely attest the execution environment to the server?

That's effectively a future without the web at all.

How would the blocker know which parts of the memory to block?

The same way trainers, aimbots, and other hacks know which addresses are critical for exploiting game mechanics, by doing analysis of decompiled code and also searching running memory for known patterns and then patching them hot.

Don't websites compile on every page load, giving them the opportunity to obfuscate?

That's a bit different than a video game that was compiled only once.