Protecting free speech is important. However most people probably agree there should be a line drawn (for example I doubt anyone would object to a group like the KKK being banned). The problem is enforcing that line fairly and not punishing people for speech that should be allowed is too difficult.
It's an interesting and important problem and becoming more so now that everyone has the means of reaching an audience online.
Gaiman has talked about this quite well: http://journal.neilgaiman.com/2008/12/why-defend-freedom-of-... The quotable is "The Law is a blunt instrument. It's not a scalpel. It's a club. If there is something you consider indefensible, and there is something you consider defensible, and the same laws can take them both out, you are going to find yourself defending the indefensible."
This precedent stood until Brandenburg v. Ohio when suddenly the Supreme Court reversed itself in order to stop the KKK from being outlawed, deciding that it was now suddenly really important for violent rhetoric to have full First Amendment protection. The KKK was actually much further over the line than the precedent-setting groups - they backed up their talk with a long and bloody history of actual violence and murder against black people. In fact some of the speech that the Supreme Court ruled was protected, such as cross-burning, was actually created as a warning to any black residents that they'd be next if they didn't leave town. Somehow the justices cared more about protecting this than they did about protecting actual, political statements that challenged the status quo.
The Wikipedia entry for cross burning indicates that it has been handled negatively in the last five years–that is, the act of cross burning was not protected as speech–and makes no mention of Brandenburg v. Ohio. That Wikipedia entry does not indicate that cross burning was a significant question at stake in the case, since the matter being decided was the constitutionality of the Ohio statute.
Admittedly, that's just a reading of Wikipedia and I have no training in law of any kind, but it doesn't seem as provocative as you put it.
Well, the ACLU would. And has. And for good reason.
There is no right not to have your feelings hurt.
(NB for people hopping here without context: that last sentence is not meant to talk about OP)
This is one of the advantages of having a written constitution. You can't just do things on a whim that violate basic principles of the country, since they're clearly codified in writing.
As for doing things on a whim that violate basic principles of the country, when it comes to constitutionally-mandated protections against unreasonable search and seizure, I'd rather go through UK airport security than US. Then there's oddities like 'free speech zones', where you can say what you like, but only where you're told to. The second amendment is hardly 'clearly codified' - it's highly ambiguous, yet has such significant impact on US society.
The US consitution has some great stuff in it and was a watershed document, but it's not a magical shield simply because it's written down on a document named 'constitution'.
Moreover, the UK would then get a lot of shit overseas, particularly in America, for being an example of how European-style socialism leads to a lack of civil liberties, and that might put pressure on the British to actually codify that right.