Can you go work for the client and capture that money for yourself?

The IRS has made that extremely difficult for software contractors, and most companies will prefer to avoid the high risk of an individual contractor being reclassified as an employee.

Usually not because large clients are worried about contractors being classified as employees.

Yes of course, but companies will prefer to work with middlemen rather than directly employing individual contractors because it makes legal and accounting easier. There’s often a lot of hurdles to employ an individual directly, like requiring the contractor to have a million dollar insurance policy say

Seems like a lot of money for that, but companies are willing to pay it

I'm a corp to corp subcontractor and aside from getting nearly half of my rate taken, I keep and pay for my own liability and e&o insurance.