Entirely depends on your child.
I have very strong political views and am an absolutely commited (and dedicated and passionate) state school teacher. Yet my daughter goes to private school, a decision we made because it was right for her.
I can't defend it on ethical grounds and I don't care. She's my child and I'd see the world go up in smoke to make sure she was ok and thriving.
Currently state schools are under funded. In a perfect world it's a decision nobody would need to make, because every child would access to an excellent free education.
But we don't live in a perfect world, and you choose your poison.
That's not to say private is necessarily right for your child. My daughter has a specific SEND but is also extremely academically able (I am not being an arrogant dick here, only saying this because it's relevant to the discussion). The small selective private through school she has gone to is actually quite capable of catering to her needs, and providing her with the challenge she needs. I strongly believe that, with the best will in the world, even an outstanding state school teacher, with 30 or so kids in the class, could not have met her needs so well - it's not humanly possible.
Having said that, she has had numerous class mates over the years with different talents and needs, that her school has woefully failed to meet, and they have gone on to thrive, both personally and academically, in the excellent local state school.
It's difficult - how can you know what will suit your kids until you try it out? So I would say keep an open mind with both options - start with the one that seems to suit them best, but keep an open mind to changing if your kids start struggling, whether that's socially or academically.
Also remember (and I wish somebody had told me this when I had to make this decision). We crucify ourselves trying to do the right thing for our kids - but we don't have a crystal ball. No "mistake" you make will be irreversible.
So do what seems best now but stay involved, stay thoughtful, stay flexible. You won't know your children's capabilities and needs fully until they are a good bit older.
And the nerdy best of luck whatever you decide!