I think this is it - he was welcomed to the club when he was new but his popularity/novelty factor has dive bombed and people get sick of people looking for freebies. Plus there's no more Sentabale polo matches and funding.
As regards William, I'm not sure. Back in the day when Prince Phillip played I think he/the Queen bred quite a lot of his polo ponies. Princess Anne's first eventer, Doublet, was bred by the Queen as a polo pony and gifted to PA, perhaps the horse grew too big to be suited to polo, he was a thoroughbred cross I believe. When Charles started being keen on polo I imagine the tradition if not of breeding their own horses, at least owning his own string continued.
Then I think he probably kept a few for William and Harry, I can recall an interview with Harry sat in the stable yard, horses in the background. (And let's not forget a certain girl groom...🤨).
Now I wonder if William just gets lent horses or uses club horses for the odd match, but I think there's a military tradition of playing and there are probably a few kicking about at Windsor for him to use. I'm pretty sure I recognise the white faced horse in the photo above, which must be a favourite of his, though of course you need several for a match. I've never seen pictures of him riding in Windsor Great park, like a certain AMW used to, and Edward and his family sometimes do. But he has to ride and practice, perhaps in a more private area or indoor school. You can't just hop on a horse and do Trooping the Colour or play in a polo match if you haven't ridden at all for several months. So he may keep a horse or two.
We don't even know if the children are learning to ride, Catherine is famously allergic to horses, but I can't imagine a royal child not learning to ride and how will George do Trooping the Colour when his time comes if he hasn't learnt?
As an aside I also noticed William doesn't wear spurs for polo. No pictures of him on a horse with bleeding sides from the spurs like there is with Harry. He seems to genuinely like his horses and always gives them a pat and checks them over before mounting I've noticed.