That's correct, but it's not just "stop paying rent", as that's illegal. In general, a tenant union would either facilitate or assist with helping the tenants set up a rent escrow to be released when the problem is solved and the rent strike ends

You could work with a court and do something like put it in escrow so the court can see you not paying rent isn't just because you can't. Instead it's about withholding money from landlord to begin negotiations. Basically still acting in good faith but also withholding money to give you leverage. IANAL

Double-check this for your state. For instance, in Ohio if your landlord is not keeping up on their end of maintaining the property you can go to the court and set up an escrow account with them that will hold your rent payments until ALL of the work is completed to YOUR standards, not theirs or the court's, and they are legally barred from taking any action against you for the duration and just about anything they do afterward is immediately suspect of being viewed as "retributive action" which is illegal under the state's renter's rights laws. Always read your rights for your state.

Not sure, but it'd be cool for an entire building to withhold rent for building repairs, or repairs for an individual.

It is legal though. They legally cannot stop unionization.

Even the US president made a big speech about being pro-union last week and that it's illegal to stop organizing them.

Well the idea is the landlord can't evict everyone in the building at once. They lose a ton of money and it's very hard to go to court that much.

Of course it isn't really like a real union. They can easily replace tenants; it's not a skilled position. Co-ops are better, since you actually own the building.