I don't think we should expect websites to sanitize escape characters. But tools such as curl perhaps should have (or already has?) some option to do it.
But this would require coordination in POSIX, in the terminals, and in programs using the terminal :-/
But the advantage would be that then you'd be able to put your terminal into this safe mode by default. And the terminal can alert you when a program is writing out control codes that are being ignored, and then you can whitelist them and/or get those programs fixed to fence off their sections of "I'm going to output data from god-knows-where now" with this new terminal mode.
It's not going to happen but we can dream ;-)