If you don’t have the right credentials then you can’t teach legally in a high school. It doesn’t matter who is paying you, if you’re there without the right pieces of paper then you are breaking the law. It’s a matter of the physical location, not the ‘employer’.
I know of school principals that have been fined large sums of money, and threatened with jail time if they re-offend, for having foreign teachers on site that were provided by a recruiter.
I also know of high schools that have hired teachers ‘informally’, without the correct paperwork and without using a recruiter.
The simple truth is that most schools have very limited understanding of the regulations and prefer to subcontract the recruiting of foreigners to agents. Even basic stuff like taxes can be a mess, as the publicly available information on hiring foreigners in contradictory. By using a recruiter they also hope to insulate themselves against the problems that usually come up when teachers complain, disappear, etc. by passing those problems on to the agent.
If you are able to present yourself to a school in such a way that they believe you to be competent and reliable, and make the paperwork simple, then you can work without an agent. It’s probably easier to go through an agent though.
I’ve done bits and pieces for a company called FIEC in the past, over a period of five years, and they have been … OK. I’ve bitched about ‘the dark lord’, but he grew into his job and we’re friends again now. Many of their teachers have been with them for years. A good way to find out if this work is for you would be their summer camps, which I’ve done a few of and always enjoyed.