Public education in Greece is in deep crisis, and it cannot be compared to the UK one: https://www.tovima.com/stories/greek-education-system-in-crisis-infrastructure-gaps-underfunding. It is not hypocrisy, as he didn't try to shut down private schools, but increase the taxation. He would be a downright fool to send his school to a public school if he has the money to send them to a private school. Furthermore, during his premiership, he demonstrated a commitment to improving public education, though his efforts were constrained by austerity measures.
The term "working people" is somewhat vague, as nearly everyone is part of the workforce. To be more specific, we need to clarify which fees we are discussing and which segments of "working people" can afford them. Let's consider a practical example: the fees at Rosemead for children Year 3-6 are approximately £15,000-£16,000 per year. For a family with two children, this amounts to around £30,000 annually, just until year 6. What kind of salaries can support these expenses? The working people I know certainly cannot afford this. While a fee increase from £15,000 to £18,000 is significant, it is not an astronomical jump. Additionally, the revenue generated from taxing private education would benefit all students, potentially improving public schools for everyone. Although this taxation might pressure public schools as some private school students may move to public schools, the majority—93%—of students already attend public schools and would benefit from the increased funding.
Left-leaning people can still send their children to private schools while supporting public education. They advocate for public education to be as good as private education. Currently, it is not. Your argument is similar to suggesting that leftists should give away all their money, which is not a valid argument.