I agree with @Pashazade! No decision is permanent and it makes sense to meet the needs of the child you see before you, rather than subjecting her to an environment which isn't right for her in hopes it will work out in the long run.
My feelings on the subject are the same as yours. Young children are driven to learn through play (not "play-based education" but actual genuine unrestricted play) and school deprives them of their full dose of play. Psychologist Peter Gray makes an excellent case for play here.
This was why I didn't send my eldest to school and used home education as a means to delay school start until a time that seemed more developmentally appropriate. I imagined she would want/need school around the age of seven and that I would send her then. It turned out I was wrong. She learned so well and was so happy without school that there never was a good enough reason to send her. We carried on HEing first her and then her younger sister, whose special needs made HE a no-brainer. Now they are 20 and 13 and it's one of the best decisions I've ever made.
My eldest did try school around the age of ten but didn't stay. In her eyes it wasn't a good use of her time. I once asked her whether she learned more while HE or at school. She said she learned similar amounts in both places, but that school took up an entire day whereas she could learn the same amount in a fraction of the time while HE. That gave her more time for hobbies, friends, relaxing and PLAYING - still very important at ten.
Though my eldest was a confident, sociable autumn-born people-pleaser, I still felt school was unlikely to meet her emotional needs. Poor staffing levels, crowding and performance targets conspire against teachers' valiant attempts to nurture children. It just isn't a good place for a 4yo, IMO.
You might need a different educational approach from the one I used, if you are definitely going to send your child to school. You'd want to ensure that her reading and writing are well underway whether or not she seems ready and willing. Schools do that from age four and they expect it, so it's likely to deal her confidence a big blow if she pitches up to school aged seven unable to read. A selective private school might not even take her if she doesn't perform as they want in their entrance tests.
But you could still give her most of the benefits of home education even if you do make some concession to the school system in the form of enforced reading and writing practice. One-to-one attention means she'll learn more effectively, so you could work on it say half an hour a day. You wouldn't necessarily have to start at the age of four but could leave it later.