Ok, gave up on that surgery. They kept insisting that my daughter's EU family permit meant she could only be in the country for six months and kept calling it a visa.
Actually the FP is multi entry permit for an non-EEA citizen family members of an EEA citizen. They can stay for up to 6 months, but their EEA citizen sponsor can't, as these can only stay 3 months under EU law, unless they become a qualified person. Other EEA country only issue FPs for 3 months, but the UK allows these non-EEA citizens 6 months on a FP, just as other visitors to the UK also get 6 months.
An FP doesn't mean they can use that member states healthcare system as a citizen of that county would, nor work, as this only come from their EEA citizen sponsor being a quaified person at all times. This is why empoyers and universities, want an RC asap. We see many who enter on a FP, work and then find they have to stop work as they can't get a UK RC.
The 2004 Directive states that EEA citizens and their FMs can only use that member states healthcare as a citizen of the country, if their EEA sponsor is working. Most EEA countries ask for proof of that the/their EEA citizen was working the previous week, or they bill them for any healthcare, or bill their own EEA country via the EHICs. Just as many EEA countries tell EEA citizens and their family members, that they must all register with their police. The UK doesn't do any of this as the UK governments have been very lax.
That said, the UK is still very generous, at present, and will let student qualified persons and self sufficient qualified persons and all their family members, use the NHS for free instead of billing them or their EEA county for their healthcare. However to be legally in the UK these must still comply with the EU Directive and all have a CSI each. I assume this is still a hangover from when the NHS was free for everyone who was legally in the UK, which ended in April 2015.? However, the UK can start to bill these/their own EEA country via their EHIC, under EU law, just as other member states already do to the British/the UK via EHICs.
Most NHS surgeries/Trust, have been very lax and it has been easy for EEA visitors to nip over and use the NHS, and the same with non-EEA visitors as we have seen on this forum. Unlike England, NHS Scotland might not see many EEA citizen and their family members and don't know what they can have and what they can't have for free. As you said, they never had one like you before.
A GP does not have to put someone on their books, as that is their choice.
You don't need RCs under EU law, but in the UK it is easier to have these to use the NHS, work, university as a Family Permit is just an multi entry permit. An RC is proof that,
at the time it was issued, the EEA citizen/their EEA citizen sponsor, was being a qualified person in the UK. However, the RC becomes invalid if the EEA citizen stops being a qualified person and they and all their family members then lose their right to reside in that member state.
Basically, it's hard in the UK to not have an RC, even though the Directive says that EEA citizens and their Direct Family Members, don't need to get these. This is because the UK is so lax in checking, compared to other EEA countries. Extended Family Members must have RCs. Although it is easy to not be caught, as the employer, the NHS, university, has no idea that their RC has become invalid, unless the holder tells them.