I see this issue differently. I put the provision in because some organizations literally are adjuncts to LE or receive referrals and/or are part of the legal justice system, and minors living here in violation of immigration laws are being protected by LE and the general criminal laws of Tennessee, mostly from the very people who brought them here.
The organizations you are referring to are providing voluntary assistance to the public at large, not under the mandate of providing a necessary service that is an adjunct to LE. There should be a high bar to receiving public funds, and if an organization refuses to be vigilant with public funds, then the organization needs to seek funding elsewhere.