With the GP practice that I am registered with in the UK, the receptionist's questions would have been in the opposite order.
If you are presenting with a new medical problem, after being asked what the problem is the receptionist (who is a trained health professional, not just a telephone operator!) would "triage" the problem and possibly transfer the phone call to one of the practice's nurses, rather than immediately take up a doctor's time with something that might not need that level of expertise.
I would add that the receptionist has access to your computerised medical records, so you don't need to describe things that are already on that record in detail - the receptionist can read!
In fact "admin" type problems (e.g. with repeat prescriptions, or test results that have gone astray for some reason) are usually sorted out by the receptionist him/herself, without anyone else having to spend time on them.
I have no personal experience of refusing to give any information about a medical problem, but I suspect the result of that refusal would be put at the "back of the queue" so far as getting an appointment was concerned.