Getting hired by a recruiting firm

Has anyone ever been hired directly by a recruiting firm for a public school position? As in you work for a public school but your contract is directly through the recruiter (private company). I am on the job hunt and was offered a position this way. Would there be any advantages, disadvantages? I turned them down and applied directly with the district instead because they’d already given me the lead and I figured the benefits would be better this way but I’m wondering if I made a mistake.