No because l believe in diversity and choice. There is no compulsory private pay as you go schooling so if you don't want to pay you have the option of tax payer funded paid for "free" schooling. This is unsurprisingly paid for by someone else and not you (apart from your tax contributions). Unsurprisingly it is likely to be funded from many of those who have more resources and so pay more tax and possibly also the same wealth creators and employers/business entrepreneurs.
In addition there are always a few examples of high achieving students from socially economically challenging demographics who will with natural intellect and ability despite the odds, come up through the ranks and say make it to Oxbridge and read a sought after and exceedingly competitive subject such as law, jurisprudence, medicine, engineering etc (amongst other challenging university subjects).
To take this concept to the very core you are essentially either wishing for uniformity where the whole population are identical ie at more or less same level in any/every relative benchmark be it intelligence, wealth or other attributes which differentiate people and creates a hierarchy/ranking etc.
Even in hardline militant communist society there is no equality. Most say in Cuba, North Korea are poor but there are massive differences right at the top and the ruling elites etc. Some are evidently more equal than others.
In life there is sadly no equality. Some one will be better than you and conversely some one will be worst. Not everyone is the entrepreneur business owner or political leader or CEO etc and someone will inevitably be the cleaner, driver, manual labourer etc.
It would be absolutely amazing if everyone was relatively equal and say prosperous, middle class and well educated. That nation I don't think exist and I personally think from my experience that Singapore, Taiwan, South Korea, Japan and Norway, Denmark, Iceland and Sweden is possibly the nearest to that ultimate utopia (for some). And even in these nations some of the less desirable low level roles are imported from overseas less developed poorer nations.
So going back to the original poster's initial proposition I do not agree that private pay yourself pay as you go schooling should be banned. There will always be a demand as they are essentially just business as say with private nurseries for babies and young pre school infants. There are always consistently highest ranked top "free" state sector schools which tend to produce a comparable and sometimes higher academic attainment. Though I suspect will not include all the other extracurricular enrichment activities. Or at least not to the same standard.
I hope this makes sense and open to be persuaded by a suitably robust rebuttal.