Correct me if I am wrong to distinguish between medical emergencies and non-emergencies. But here I'll focus on NON EMERGENCIES here Undeniably, physicians can refuse patients with good reasons — "If the waiting times for a service are too long you may be told about different places you can get the same or similar treatment." "Each NHS trust can decide which types of behaviour could lead to treatment being withheld or withdrawn and how such policies are implemented."
In the NHS, you must book a consultation with a physician — 1. your Offer. The physician or her clerk must accept your officer — 2. Acceptance.
Your consultation causes the NHS to pay the physician — 3.1. Patient's Consideration to Physician. The physician is supposed to assist the Patient — 3.2. Physician's Consideration to Patient. Furthermore, under the Contracts (Rights of Third Parties) Act 1999, s 1, patients can enforce the contract between the NHS and physician because this contract undeniably "purports to confer a benefit on" patients.
4. Intention to create legal relations is obvious — GMC guidance is clear: “Wherever possible avoid providing medical care to yourself or anyone with whom you have a close personal relationship.”
5. Certainty of Terms isn't an issue, if you frame it broadly as a Patient's receiving medical assistance or advice from a Physician. Thus
why doesn't Contract Law alone cover fully Medical Malpractice? I've perused three hornbooks on English Contract Law, and none bring up Medical Malpractice.
Why does Medical Malpractice involve Tort Law at all? They feature in at least 2 English Tort Law hornbooks. See Markesinis and Deakin's Tort Law (2019 8 ed), p 902.
See also Christian Witting, Street on Torts (2021 16 ed), p 700.