Hi OP, my adult son has a severe learning disability and you are right the system can be very confusing and not really joined up. The first thing to understand that there is no such thing as being registered disabled in the UK. You have to navigate each bit of the system separately, but it is the case that being recognised as disabled in some parts of the system helps with getting things sorted elsewhere. And how it works depends on the individual person's needs and capacity to make their own decisions.
My son's disability is so severe that there really is never any argument about his capacity (genuinely the only decision he can be trusted with is what flavour ice cream he wants). You describe your son as "vulnerable", and I understand that must be a huge worry for you, particularly if he doesn't want you making decisions for him. But if your child receives DLA now, you will want to apply for PIP to start at age 16 and that's a good place to make some of these questions more clear.
As part of the application process you will have the opportunity to apply to be your child's nominee. That usually involves someone coming to your house and meeting your son and assessing their understanding of money and ability to manage it. If they approve you as nominee, the PIP will be paid into a bank account that you have access to, and it will be your responsibility to manage, but it will be your son's money. It's sensible to set up a separate account to avoid confusion.
And being a PIP nominee does seem to set the tone for all the other things that need to be sorted. However, if your son isn't eligible for PIP, then essentially the system is concluding that he has capacity to manage his own affairs. Being vulnerable (which all 16-year olds are tbh) is not the same as lacking mental capacity.
The medical side of things has always been quite easy for us, but my son is generally physically quite healthy, so the conversations are not particularly complicated. Because I am his PIP nominee it was quite straightforward to be listed as his appointee at the GP and in his hospital record. As PP have said, I can't get access to his NHS app on my phone, but I can login to his account on a PC browser.
Good luck, try not to get overwhelmed with the whole big system. Start with PIP and take it piece by piece from there.
Edited to correct typos