1) They don't need to pay to advertise.
2) They don't have the time or the inclination to sift through lots of potential applicants
3) They cannot get lots of potential applicants
4) They want the contract to be between the school and the agency rather than the FT
5) They can dump the FT when he clouts a student due to the fact he didn't take his meds, then demand a replacement FT quickly.

The main reason is number 5 of your list. The dumping could be because of a variety of reasons. The procurement of a fast replacement is essential in the eyes of that school's headmaster. Once an FT is in a school's bad books, that school will want him out the door ASAP!

Number 3 will also be a major reason in my view. Perhaps a school got only a couple of applicants by advertising direct. From a hiring perspective, that's not a good situation.

Better leave it at that for Turnoi's sake. Glad you've found a couple of decent recruiters. They aren't all disreputable despite what some posters claim!