The fact that CAWHT is the only campaign to have “shown any capacity to involve and mobilise people on an ongoing basis” owes more to the moribund nature of much of the Left than anything about the strengths of the CAHWT itself.
The resources of such parties as the SP and SWP are limited, and any issue that they choose to campaign on means there is an opportunity cost in terms of other issues. That is why even if, yes, you do put resources into the TUs, the CAHWT represents a lost opportunity of further still resources.
As for an alternative, it seems clear to me that a strike by workers will strike the ruling class where it hurts in a way that merely withholding payment of some money which will just be got elsewhere. Look at the poll tax revolt in Britain to which many attribute Margaret Thatcher’s downfall. The Tories just replaced her and carried on in power, and it’s very difficult to imagine even that eventuating from the CAWHT.