As a resident of the UK, I think that if we did have a written constitution, regrettably free speech wouldn't be on it. As much as I personally believe in free speech, I do get the impression that the general public here don't.
Which is absolutely fine ― at least then those people wouldn't be able to hypocritically claim free speech as one of their country's positive attributes when trying to distinguish themselves from "barbaric" countries like China or those in the Middle East while also suppressing free speech when it fits their "multicultural" agenda.
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.
I had a British lawyer tell me that in no uncertain terms, the UK has a written constitution. It's just not on a single document named "The Constitution", but a collection of documents covering different things.
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'.