Is this a legal loophole I can exploit to get out of a non-compete agreement?

Of course you should be grateful to have a job. Don't take my word for it, just ask the people you work for.

I've made this switch a number of times, but never looked for any legal loopholes. In fact, I didn't look to make the switch at all. The company I was working with could also do the math, so they worked things out with the employment agency. Employment agencies routinely let temps out of their non-competes at the request of the employer, because that's good business for them. However, companies can also have legal, tax, and other reasons for preferring to work with freelancers through an employment agency, rather than directly. If the ad agency hasn't expressed an interested in working directly with you by now, it could be that this is because they don't want to. If they do, then they would be the ones to resolve this with the employment agency, not you.

If you go in a legal way, then you will not face any kind of problem in future. You totally want to stop the middle man process, then you can direct consult with them, no need to talk with the middle man, make sure that the way you have chosen it should be in a completely legal way. Then I am sure you will not face any kind of problem.