To be fair, some are more democratic than others.

However, because the private firm is not legally, or practically democratic at all, it makes sense for those who love democracy (and themselves) to unionize, yes.

There are multiple democratic institutions I belong to where I don't generally feel like I have the power to change things.

Democracy is great, don't get me wrong. But it doesn't magically mean that representation for your specific subgroup will exist. If you're part of a demographic with 10-15% representation in your union, is it really a crazy idea that your demographic might want to pool its votes together to ensure that you have more collective influence in a process that might otherwise ignore you?

Collective, coordinated negotiation being more effective than individual negotiation is the entire point of unions. It's just not clear to me why those principles would stop applying within a union itself.

To be honest those principles shouldn't stop within the union. But that is really the challenge for every movement with similar goals. The maoist idea of "continuous revolution" tries to address this problem.

lol. i guess you've never actually been in a union.

I've been in, am currently in, and have been involved in unionizing workplaces.

Companies are extremely motivated to undermine unions. Unions tend to get stale as they age and lose their militancy. That's why wildcat action is so common. It sucks but even a workplace with a stale union is better off than one without collective bargaining.

The hardest part of organizing, like so many things, is going from 0 to 1. Once you've got a union you have a lot more power. Thinking you don't is what the company wants you to think.