Meh. I had blood tests every other week from the age of about five. No freezing sprays back then. Or specialists in Paediatric Phlebotomy. For the kids who are needing blood tests anyway, it's no hassle to take another phial. And when they don't, it's not that big a deal anyhow (probably useful to get them accustomed to the procedure for the future anyhow).
Mind you, I wasn't the type of kid that had the screaming adabs over vaccinations, either. I was shocked one year at work when I saw the setup for the HPV vaccinations - the school staff had laid out PE mats and blankets for the numbers who would 'faint' from the sheer agony of a tiny needle in their arm - apparently, once one went down, everybody behind that one would follow. It's a fashion. Nobody gave a fuck about having the things in the 70s and 80s apart from the one or two 'delicate' children and mad antivaxxer parents like my FIL .
A small child is capable of saying yes to a blood test. They might not be able to quiz you about funding and the declared interests, but they can say OK to doing it to help doctors to find out ways of making sure other children don't get sick. By the time I was six, I'd probably be asking what diseases, haven't they been vaccinated and lecturing Phlebotomists on the correct gauge needle and showing them the most productive (and least painful) spot to get blood from.
If you're all flappy over vaccinations as per your post about not being sure if your kids would have the second one, you're not the type of parent they want to hear from, so just chuck the information in the bin and leave altruism to others.