My oldest's were delayed and he had his at a Children's Hospital, and was under observation in the open play area they had for I think it was about an hour to see if there would be any issues. I was told at the time that any significant issue would most likely show something in that time frame. He just played for the hour with occasional checks, thankfully no issues, and is now adult.
With your concerns, I would ask if there is an option in your area to have them at a hospital with support along. I'd be surprised if there isn't some option to have something similar if only to ease parents' minds enough to help encourage increased take-up.
I get being worried. I was too. It was nothing to do with conspiracy theories or autism for me - it was having been told by family members how terribly ill my sister was after her jabs, having a child I was already struggling to get help for, and being a disabled person who had already experienced medical professionals verbally, physically, and sexually abusing me before I was a parent, and then more being really shitty and blame me for anything being less than ideal in my pregnancy, childbirth, and my child's early days. So yeah, I struggled to trust medical professionals, struggled to be in a room with one without another adult for years.
Oddly, people telling me to just trust doctors and I must want to a dead kid didn't help - I'd already had a medical professional tell me I must not love my child while pregnant for being dehydrated. It just ended up as shitty people noise to me.
What helped was having my concerns and that some medical professionals are really shitty acknowledged with compassion, being treated as a competent adult and discussing the benefits and risks in a 'this is how they work, this is how they can go wrong, these are the signs of things going wrong, here's what can be done if they show up' around the vaccines as well as the diseases and support at the hospital so I wasn't alone and could talk through things as needed. That was many years ago, and it still sadly surprises me how rare compassion is for worried mothers, knowing how not unusual what I went through is.