In a discussion of NHS vs private practice, the fact that some (certainly
not most) NHS-employed doctors also do private work, has no baring on the incentives placed upon them within their NHS employment.
I am certainly not saying the NHS system is perfect, and all private medicine is bad, so if that is the impression you got then perhaps I was unclear.
In almost every single clinical situations within an NHS hospital that I have seen or can think of, the clinicians treating you will not receive any extra money, bonus, promotion, etc. for the treatment they provide you or do not provide you.
I have lived in countries with more private systems, where there is a direct relationship between what treatment/investigations you receive and what money ends up in the doctors pocket. For example, a private doctor in HK referring for not-strictly-necessary-but-ultimatelhy-clinically-justifiable MRI scans at a private MRI clinic which, provides a percentage of the (expensive) scanning fee back to the referring clinician. Contrast that in the NHS, no one gets given money for sending a patient for an MRI scan, and in fact, if it isn't going to change the management of the patient, the request is likely to be refused to constrain resources. In the HK system, neither the doctor nor the MRI provider is incentivised to not do the scan, quite the opposite.
The NHS system is not perfect, but pointing out different the pros and cons of different incentivisation structures that do have an impact on patients is not something that is a "meme that needs to die"…