Since it's a free market - why shouldn't a group of people work together to improve their lot in it?

A free market is, by definition, free of collusion.

People with large capital don't need to collude - they already have tons of power as individuals :)

It's like the term "free market" is a thought terminating cliche or something.

The term was coined to refer to an absence of gatekeepers charging for access, though collusion is often pretty similar.

There are already systems in place in companies for you to do that. If you want to bring Unions in to the picture it means

1. Your opinions are not useful or valuable.

2. The company is poor at adapting good ideas through existing channels.

In either case you should leave the company and find another one.

Why should I be compelled to use the systems someone else designed? Who is so wise and righteous that their external control over my decision to join or not is justified?

Because the system is designed mostly by founders who built the company from ground up and had the maximum skin in the game. Employees have minimum skin in the game, never have full context on things and are easily replaceable. You are always free to start your own company according to your own rules.

A free market implies a freedom to unionize and strike.

Sure. Also for the employer to fire employees for doing that. So doing it is stupidity instead of just finding a better company that values your skill.

Well no, because organizing is legally protected.

Keep in mind these legal protections actually made unions weaker, so be careful about what you wish for.

Hell yeah.

Or just take over the company. That is easier.