Vote for Oundle from what you are looking for. Went there, highly recommend, not much wrong with it but happy to outline where it might not be great, certainly not a utopia (nowehere is lol). Full boarding 850 pupils and 250 day students. No in-between weekly or hopping home for a night.
Very popular school so I would get in touch now, they are probably full but can put you on waitlist if someone drops out of going there although with VAT hike Im sure all these schools will have more places than usual.
Fully endorse others saying choose a full boarding school. Any school with less than 50% full boarding can become fractured socially (weekends with some friends not there are lonely). Most have this weekly boarding model now off the top of my head some Co-Eds are Wellington, Oakham, Charterhouse, Teddies, Bradfield, Canford. Differing percentage of full boarders, others can confirm numbers.
Oundle and also Marlborough great choices (also v popular). They a basically the same. Oundle has more day students but anyone boarding has to stay there and with 850 full it feels full boarding, the day students have to walk to the boarding school drum (some come in on weekends), never leave unless it is an exeat or holiday even if you live 20 mins away (which I did). Doesn't preclude parents seeing you on weekends for a lunch or after sport for a trip to pub/coffee shop but never nights away. Students from both are individual, driven and ready for modern world in my experience. Both quite monied, Oundle is much more understated though.
Uppingham is also full boarding one day house and quite good.
Rugby also full boarding and in Oundle and Marlborough bracket, town is a lot worse though (not sure its that relevant though truthfully, everyone who I know who went loved it and didn't think it mattered).
Good luck.