Strictly speaking they should ask for proof of identity and proof of address.
In their contract to provide NHS services there is a map that outlines exactly where the surgery will take patients from. If they choose to take people outside their catchment, and, for example, a new development goes in within their boundaries, they are still obligated to take anyone from the new development, so it is in their best interest to stick to their catchment.
Most GPs are paid per head with a formula that accounts for age, sex, etc. of the patients on their list. It's VERY rare, and a big old contractual hassle for everyone involved to close their list - they don't want to do it and the PCT don't want them to do it. IF they do, the PCT has an obligation to make sure that patients within their catchment have another option - a surgery near by who will need to agree to take the patients in the interim, for example.
If it's just a case of registering in the correct catchment and surgeries never say they won't accept new patients, then it seems unnecessary to require a utility bill as proof of address. There is very little incentive for people to lie to a GP about their address, so why not just take them at their word?
The girls are right - there is very often a patient preference. I know one area with three surgeries relatively near each other. One is shiny and pretty with loads of parking. All the GPs at all the surgeries are the same and the service they provide is the same, but given the choice most patients would fill the fancier one which means the smaller ones would lose a lot of money.
It's not like a bank account where there's the potential for crime--everyone is entitled to medical care, and not everyone is able to provide "proof" of address. If they won't accept a signed letter from the OP's landlord, and that's all the proof she can offer, they are effectively denying her medical care with their senseless bureaucratic pedantry
There are means of doing it. And a decent surgery will work with people to get them registered but they also have to follow the rules. Most people, within a week of moving in somewhere get something - a council tax statement, an energy provider bill, etc. And if there's a circumstance like homelessness that means someone actually physically cannot provide proof of address, the PCT will work with the surgery to come up with a solution like using the surgery's address as a temporary measure.