Whether he's managed 5 minutes or 10 years is not the point so I'm not sure how that is relevant in the slightest. All coaches usually have a set style of play that they stick to and it's a reasonable argument that Carroll doesn't look suitable to it.
He's the manager and he's got every right to ditch a player he doesn't want to work with.
I don't think it'll work for Rodgers at Liverpool for a number of reasons like, but you cannot critisise him for getting rid of any of the current crop. To coin a phrase from Stu in the betting thread, "don't chase your losses", they keep on banging on about the price tag as a reason he shouldn't be let loose but that isn't going to make him perform any better, they need to forget about the £35m and do what's best now for the manager and the system.