That isn’t a cartel and there can be multiple, competing unions working for the same type of workers in the same industry.
That's the definition of cartel. Unions exist to maximize the amount they extract from buyers of labor. This is rent seeking plain and simple.
What's more, they don't compete with each other, which is what corporations must do. Why does the UAW get to enjoy a monopoly on the sale of autoworker labor? Should Ford and GM and Chrysler be able to unionize together to keep wage costs lower?
Why not? If you go talk to your manager and tell him "I have another offer at XXX, I want you to match it or I'm leaving" what is going to happen?
In the past, I've been quite open when I thought that I needed more money to my manager, and have even given specific ways of making me "not distracted by money concerns". Sometimes they can meet those goals, sometimes they can't.
Personally, the offer as you've given it is probably more adversarial than I'd prefer. Something like "I feel like I'm worth more to the company than X, I feel like I'm worth Y, and here is a list of reasons, here is my career goals, etc etc". Then if they don't match it, you can accept that other offer. But YMMV.