I understand somewhat on the school one, there is an increased risk with so many kids in one place. Similarish with childcare. Not sure on the child benefit - that comes in before vaccinations and something about that feels off to me. All of them though are punishing the child for their parents choices, but that one particularly to me gives me pause.
It is interesting how different countries handle the issue. Growing up in the US, I had all my jabs, and when I got into university, I didn't have access to my vaccination card - my mother had it, but had no custody & wasn't talking to me at that point - so to confirm my acceptance, I had to get many of the vaccines again. In the UK, I found out at my DDs teen boosters that one of their childhood jabs was missing and were just told for them to make an appointment for it when they are 19. Even though one of them now works in a school as an apprentice TA, no one has really cared about her vaccines.
You say 'the medical community has lost trust' etc. and 'they need to work hard to regain that trust'. This is sheer bollocks. The science of vaccination is very well established and proven.
IME, the discussions around trust has little to do with the vaccines themselves. Some of it is lack of understanding on how they and the components in them work, but a lot of this comes from the experiences elsewhere with medical professionals. There is a reason why women during pregnancy and shortly after giving birth are drawn into anti-vaxing at a significant rate. They are one of the groups most at risk for experiencing medical abuse, and are commonly blamed and dismissed for any medical abuse experienced. This transfers to the concern that if something goes wrong with the vaccines, that they will get the blame, they will get dismissed, they will once again be out of control and at the mercy of an institution that has already shown that 'do no harm' doesn't apply to them.
While it's popular on these types of threads and other media to just dismiss these people as stupid, it's rarely helpful, only further supports the dismissiveness they've already experienced, and I've yet to see it change anyone's mind. Acknowledging medical abuse, the impact it has and how it leads to legitimate concerns, where needed discussing the science and processes with respect, and offering alternatives like having the jabs at a location with more observation or alternative schedules like 1 jab at a time or greater spacing. These I've seen move people from anti-vax to catching up and willingly paying for optional vaccines.