On the otherhand, having a plethora of unions at a workplace weakens labor's bargaining power: A US union, by the nature of having a closed shop, has the power to destroy the business with a prolonged strike. Nobody wants that (capital's out their investment, labor's out their jobs), but it serves as the nuclear option from which the union's bargaining power is derived.
I think you are confusing terms. Closed (only members can even apply to work) shops were outlawed in the us in 1947.
24 states allow Union shops (workers must join the union after hire).
In theory multiple unions could strike together. Alternatively, dissenting members of a single union can keep working.
In both systems striking is the union leverage.
This makes US unions very all or nothing, and excessive demands from unions (particularly RE: pensions) have driven many businesses out of business and bankrupted many municipalities.
My understanding is German and Scandinavian unions are much more collaborative and are unlikely to force demands that bankrupt companies.
This is why unions have such a negative connotation in the US. A lot of people see municipal governments being held hostage by unions over pensions and early retirement, and union employees being so hard to fire that you end up with "rubber rooms" and end up associating unions with expensive enshittification of services.