I think that’s correct, but that “full employment” is doing almost all the work here even while Konczal’s emotional emphasis seems to be on bargaining power. After all, if you have strong labor unions and a government that doesn’t fight for full employment, then what happens is the unions use their bargaining power to cut insider/outsider deals at the expense of the unemployed. One of the great virtues of American unions in their heyday is that they used their political muscle to push the government to fight for full employment, which was excellent and it’s a political voice we’re desperately missing today. But that’s not to say that the unions themselves are a viable substitute for full employment. A market economy is either going to operate near full employment, or else people will only share in its benefits thanks to handouts. That’s true for any given set of labor market institutions.
Sure, full employment is doing most of the work here. But that's the point of a strong labor movement: it forces the government to fight for full employment. It fights for lots of other stuff too, and that's the whole virtue of organized labor. It's true that they also produce a modest wage premium for their own members, but if that's all they did then I wouldn't care much about them and neither would most other liberals.
Unions have lots of pathologies: they can get entranced by implementing insane work rules, they can get co-opted by other political actors, and they can end up fighting progress on social issues, just to name a few. But they fight for economic egalitarianism, and they're the only institution in history that's ever done that successfully on a sustained basis. That's what makes them so indispensable to liberalism and that's what makes them the sworn enemies of conservatism.
You just can't pull labor and full employment apart. It's not a matter of emphasis. A country without a strong labor movement is almost inevitably one in which economic and political power is overwhelmingly on the side of business interests and rich people, and that means you're not going to have sustained full employment because that's not what business interests and rich people want. It's all about power, baby, power.