The actual "competition" wouldn't be among which union the employees joined - rather they would compete to undercut each other on contract negotiation, which would only bolster the employer's hand.
There are good reasons why this is not common practice - but there is also no law preventing multiple unions.
If there are no competing unions and you're in union C, then you've got no options but to quit or try and topple the union (impossible, effectively).
This depends, of course, on how exactly the new unions are allowed to form. It seems unlikely that the NLRB would implement policy in a way that allows dozens of unions, each with a handful of workers; but I guess if that happened, it would end up just like what you're saying.
The idea is that different kinds of workers could be in different unions. A department store might have, for example, cosmetics specialists, tailors, accounting staff and so on; and these groups are distinct enough, from the NLRB's perspective, to have their own unions.
Imagine a world with two or three unions per employer and you could actually pick which one you were under. They’d actually have incentives to deliver!
But we’ll never have that because it means less lobbying money and smaller campaign contributions to our representatives in government.