Yeah, technically "safe" and "you're in for a bad ride" can be one and the same.
The NHS gives you what they have available, I'd assume, and what they got the best bargain on - or could get at all. I would not be at all comfortable with a situation where they said they'd "give me the vaccine that was the best for me". Not that all medical professionals are... ill informed... but there are enough of them (in both our countries) that I would ~never~ risk that unless we were in a situation where it was a grave medical emergency and had to take a leap of faith because there was no other option!
The NHS (and a lot of docs here) also used to say that standing six feet away from someone would protect you from catching Covid. (Or was that 3 feet - or the equivalent - in England, while in Scotland it was six feet?) And that wearing a cloth mask would protect you.
I was told the name of the flu vaccine and was given a leaflet from the makers, but threw it away. It really doesn't matter to me what flu vaccine I am given and I felt the same way with the Covid vaccine: I'm just grateful to live in a country where these vaccines are available.
They had two rooms, one for the vaccine for the older people and a guy who said he hadn't booked but was diabetic, used that room too. The woman who looked in her 30s and who had to pay before she could have the vaccine, went in the room where the other vaccines were given.
The staff were very organised and also found that my change to a new GP surgery the previous evening, was not updated on the records yet. I still had my council tax bill and my driving licence in my bag that I had used to join the new surgery but they didn't need those as they could see that I was "ordinary resident" in England, for bill free use of NHS England.