There is definitely a view that private education shouldn't exist, driven a political philosophy that says you achieve social mobility by putting everyone in the same school. That started with comprehensives and when it didn't work universities were blamed for not accepting enough state school pupils, when it was as much to do with poor teaching. If we had a successful state school system there might be a case for VAT, but we don't.
There is definitely a view that private education shouldn't exist, driven a political philosophy that says you achieve social mobility by putting everyone in the same school.