It's not that simple though. Mum of two children with autism here. Both in mainstream.
To start with, a diagnosis of autism doesn't mean much, it can be a very wide spectrum so no, not every child with autism will need a specialist provision.
Secondly, you can't just chose to put you Child in a specialist school, unless you private that is. For a child to get a specialist school they need a statement/ehcp, these are notoriously difficult to get, and even then, they have to state they need that provision. So one of mine has a statement and his states he gets full time one to one support, my other child is going through the process and ideally I want it to state he will remain in mainstream but in a learning needs base. We shall see though.
Even with that statement/echp naming it there are waiting lists. That is if he gets one, figures are like 3% of children with ALN having one. Last I checked, possibly lower these days. It's a long and time consuming process and if you have to fight it can be financially very expensive.
That said, does he need that sort of support? Schools are massively underfunded and it's getting worse, but they have a duty of care to provide support for each child. If they don't feel they can meet it then they should start the process.
I think it's usually better for most children to remain mainstream, it's very unique to each child and family though. I don't know how my eldest will survive at secondary school, that may present with major issues which will mean he will need a specialist provision but I hope not because he does learn and while things take more time and can be far more difficult for him than other at the end of the days when he is in that environment he will get there, just slower and in his own way.
Its depends a lot on the relationship with the school too. My past school was horrific. They kept saying they didn't have the money and couldn't cope etc etc, which is partly true, but we moved to another school who just had a completely different attitude and it's been life night to day. If this school said they couldn't cope I would immediately listen because I know they are interested in the best interests of the children.