3+ 4+ boys assessments Northwood any advice greatly appreciated

[Jiggleberry]

Hi all
Looking for some guidance from veteran mums/dads.
My son attended 3, 3+ assessments for local independent boys schools and did not receive an offer from any.
I was very laid back about the whole procedure (in retrospect, too laid back). He is a fun loving bright boy. I didn't put any pressure on him about these assessments and told him we were just seeing what toys these big boy schools had.
1 went horrendously as he was overtired. 1 I thought went well, he detached from me easily, played and shared nicely and willingly went into the class room without me and came out beaming (granted I can't know how he interacted in the classroom) and the final one was so-so. They asked questions of which he answered all correctly but he wasn't great at the small things, saying hello and bye and he got upset towards the end when someone said bye to him, got upset when a teacher asked him what he was playing with. He can be temperamental like this sometimes, (it was approaching lunch time) but I have thought this is normal 3yr old behaviour but maybe I have been naive?!
I am surprised at how disappointed I feel for him and in myself.
He currently attends a day nursery and I have decided to move him to a Montessori as I think this will provide some boundaries and discipline he would benefit from.
I will retry him next year at 4+ but I would be grateful if people can tell me what to expect at a 4+ assessment so he and I can be better prepared. Does anyone have any views on a tutor and if a tutor is not necessary what can I do at home with him?
Does anyone know how many places are available at 4+ for schools such as St martins/MT Prep/ St Johns so I can be realistic?
If anyone has children who have been successful in this process I'd be grateful for your knowledge.
Many thanks!