What is your take on vaccination hesitancy?

I am part of the debate team at school and our topic was vaccinations. But the format of the debate was different. It wasn’t really a debate. Instead one team had half an hour to come up with a speech in favour of vaccination hesitancy, and after the speech was through the other team would pull arguments from that speech, and so the debate would commence. One of the boys on the other team (the team with the speech) apparently came up with this concept that I will mention ahead, and the team began their speech with this (not exact wording just the gist of it)

Imagine a future world where everyone upon birth is injected to alter their bodies’ natural buildup against diseases. In some cases it works to make them stronger. In other cases, especially in third world countries, it may be mixed with additives that can harm children. In some areas children suffer from sterility later in life due to those injections. Yet parents are forced to inject the child or face severe consequences.
Sound familiar? Because this is no dystopian world. This is today.

And this got me thinking. I live in a third world country. And I have personally seen these things happen to children after vaccinations. My cousin has been fully vaccinated yet is constantly sick, almost as though she has no immunity. An uncle of mine is sterile and the batch of vaccines which people his age were injected with is the only thing we can find to blame since three of his childhood classmates that he is in touch with are also sterile and none of them have suffered from any serious childhood disease that could’ve left them sterile (like the mumps) or had any such accidents.
And this has me wondering. Especially since one of my older cousins is pregnant. And she isn’t sure if she wants to vaccinate her child. This is because we have auto-immune disorders in the family. Two close relatives have suffered from ALS (the baby’s grandmother and aunt) and my cousin is worried that injecting her baby so soon after birth may not give the baby proper time to develop a good immunity and she’s worried that exposing an immunity that is already at risk of auto-immune disorders to diseases before it can mature may increase the baby’s chance of getting an auto-immune disorder. She says she will get the baby vaccinated when it is older and has had time to develop a proper immunity.

So I get that there are a couple questions within this question so I’ll separate them.

What is your take on vaccination hesitancy?

What is your take on vaccination hesitancy in third world countries keeping in mind problems that seem to arise from vaccinations?

And is the science of my cousin’s thinking right and should she avoid vaccinating her baby early?