A system with school voucher should solve the problem with profits in welfare. Give a school voucher to the pupil/child guardian. If the fee of the school is lower than the allowance of the school voucher then the pupil can choose to pay less and save the difference for future education. If the fee of the school is higher than the voucher the pupil can value the offer and pay more than the allowance.
There will be schools with special education for special pupils and these schools will be given subsidies in order to be compensated for higher cost.
This very simple system will eliminate the problems with profits in welfare.