Well at three they still need fairly consistent supervision, meals making, possibly help the loo or nappy changing, help getting dressed, taking anywhere etc. How is that not full time work?
Honestly speaking, if you find it hard work to look after one NT 3yo, perhaps you're not suited to being a SAHP. I do understand that some people find it hard to be around young children, but if that's the case, it might be better to go back to work.
As to whether a nanny in charge of a single 3yo is working, well, obviously, yes they are, but as jobs go, it's not exactly the hardest job in the world. That said, it's very different from being a SAHP. For a start, it's much harder looking after someone else's child than your own. Secondly, as a nanny, you are required to meet the expectations of your boss, which as a SAHP, you don't have to do.
If the SAHP refuses to take on a greater share of the housework/admin, I can't really see the point of having a parent stay at home - unless their earning potential is very low and childcare would cost more than their wages. I certainly wouldn't agree to being the WOHP in that scenario!
Obviously, it's different for families who have multiple young children or children with SN, where the childcare will represent a much greater workload. But one NT pre-school child and an NT 6yo don't represent a full time job imo.