I'm both the parent of a SEND child, and a childcare professional, who works predominantly in the 2-3 room. Speaking from a professional perspective, we know more about child development than you do. You need to listen to the professionals. If two separate professionals have expressed expressed this concern, there is an issue. Long term outcomes are so much better when children can access early intervention.
Speaking as a SEND parent , you really need to get the ball rolling. Waiting lists are insane. My daughter is 10, her previous schools refused to acknowledge her difficulties, and sent me to parenting classes, but her current school approached me with concerns on week 2 of her being there. It took us 8 months to see an educational psychologist, 10 months to see the neurodevelopment team, 23 months to access occupational therapy and 28 months to get a speech and language assessment. We're currently 2 years into a 5 year wait for an ASD assessment.
Even without the ASD diagnosis DD has 7(?!) diagnosed Neurodevelopmental conditions, some of which I never even knew existed and wouldn't have known the signs of. She will be going to specialist school in September. And yes, she can maintain eye contact, follow instructions and respond to conversations.
Everything has been an absolute battle, from getting underfunded overstaffed schools to acknowledge her struggles and give targeted support, to getting an EHCP and getting the LA to see that she can't cope in a mainstream school. The older they get the harder it gets. Take the opportunity to access early intervention.
You should speak to your HV and request she do an M-CHAT assessment, this is the modified checklist for autism in toddlers. If this comes back clear I would speak to the GP and request a referral to the community paediatrics team to assess for a developmental delay. Provide the letters/emails confirming that your son has been excluded from two childcare settings because they cannot meet his needs, so the GP can send them with the referral.
I know you are against the idea of your son being autistic, but waiting lists for assessments are so long that if the toddler screening comes back positive, I'd consent to a referral because that way he's in the system and if he suddenly outgrows his behaviours you can just ask for him to be removed from the waitlist. It's better than the alternative of things getting worse and not getting an assessment until he's halfway through primary school.
Your son sounds very sensory seeking from your description. I've attached some suggestions of ways to help fill those sensory needs, which may help his behaviour and stop his pinching and grabbing etc.