Technically this is true. If you make a statement to the police, then that statement can be used as evidence at trial (both against you and in your defence). If you don’t make a statement, then the only way to get your testimony in front of the jury (should that be necessary), is to take the stand, where you will be subjected to cross examination. A process where you (probably not a lawyer) will have to argue with a lawyer who will be doing their best to make you trip over every single thing you say, in front of a jury, for a number of hours.
So while it does potentially have a benefit, it’s still always a bad idea, because you’ll want to see the discovery before you say anything at all.
Your explanation for the UK police caution phrasing makes perfect sense, but I wonder now if it's not meant to scare people into talking?
I don’t know the motives of putting that in the warning. But it is at least true, even if it’s almost universally not a good idea.
http://news.bbc.co.uk/onthisday/hi/dates/stories/october/20/...
I do not know what edge cases the addition of "But, it may harm your defence if you do not mention when questioned something which you later rely on in court." was meant to protect against, and how, but it is absolutely nowhere near obstruction of justice or contempt of court.