Think of it like a stop-and-search power. It is too wide, but not choosing to stop and search everyone who they have the power to isn't really like selective enforcement - it's just a power they can choose to use if they feel they need to.
I'm not very familiar with US law, but a few moments googling suggests that that's nonsense, and the standard the police must reach to conduct a stop and search in the US, for e.g. weapons ("reasonable suspicion", per Terry v Ohio), is essentially identical to that in the UK ("reasonable grounds for suspecting"). Though the UK statute does cover a slightly wider class of items - e.g. stolen property, rather than just guns as in the US.
(It's true that the reasonable ground requirement was removed for certain areas by the Terrorism Act 2000. That provision was held incompatible with Article 8 of the ECHR (our nearest equivalent of your 4th amendment), and has consequently been repealed).
I'd also raise an eyebrow at your implication that the police are generally less prone to misuse of their powers in the US than the UK. I haven't researched it, but my impression was that in practice it's rather the other way round.
The ultra fragmented nature of the police forces in the states lends its self to this unfortunately.
What the USA needs to do is to move into the 20th century and for each state to have a single police force with uniform standards - this would save a lot of resource wasted by every city/town having its own police force plus state troopers, sheriffs and what have you
