I think a few people are overthinking this. We currently have a good amount of capacity for vaccinations. There is a dual system for booking, you get it either through your GP or the NHS site.
It really doesn't matter which method you go for, they both get entered in the same database. When the government announced that group 4 could book, then literally anyone eligible is welcome to use the website.
I am a GP. If a lot of my patients use the NHS website, it just means that we get through the current group quicker and down to the next group. Next weekend the government will probably announce that group 5 can book and the NHS website will allow this. Our practice has already moved on to group 5 but there are some areas where this hasn't happened yet. In those areas booking via the NHS website helps balance things out.
You are not stealing an appointment from anyone, once your group is eligible it is eligible and the sooner the group gets done the sooner we can move down to the next group. All that matters is getting jabs into arms. We know that Royal Mail has been struggling. We also know that the NHS is super busy so generating letters, putting them in envelopes and posting them takes time. There is no point wasting time here, we need to just get on and vaccinate.