My son (who is and will remain in public school, though we live in a wealthy suburb and so enjoy de facto private schooling) had trouble socializing in 1st grade and was "diagnosed" by a school functionary as having an "autism spectrum disorder". After consulting with my mom, a full-time LD/BD teacher in the Chicago Public Schools, we argued for over a month against assigning an IEP to him. "You need the IEP to get occupational therapy services" (read: handwriting lessons). Yeah, and, we can count on never getting rid of the IEP, or getting him into any selective program anywhere.
(He's concluding 5th grade now, and besides issues getting his homework in, he's doing great. Late bloomer.)
The fact that private schools can be selective about students and can expel them virtually without cause is not a spurious concern. It's a very real issue.
Mentally handicapped, ESL, and problem students are easily booted from or denied entry to private schools.