I can answer what I know as a Dentistry student.
On the NHS, you get paid based on the amount of UDAs you complete. UDAs are awarded based on what band of treatment the dentist completes (1, 2, or 3), you can find what treatment is included in which band online. Each UDA is worth a different amount based on the location and what you agree with the practice owner prior to starting work there.
NHS dentistry is very stressful for Dentistry, mainly due to the time constraints. For eg, on the NHS a patient who needs 3 fillings will pay the same as for 1 filling, even though it takes x3 the time. Obviously this means the dentist also gets paid the same for both work. This means the dentist may have to 'rush' (although they shouldn't do anything that may harm you). I saw the true scale of the 'rush' when I did work ex; the dentist was working like a machine. They have to meet their UDA contract targets or they may be fined (or not get a good UDA contract next time).
It's completely different in private dentistry though. The dentist charges as much as they wish, purely because they have much more time and can spend more money on the materials they use on you - which equates to you paying more for the private service. There is no bands and UDAs, so 3 fillings would cost more than 1 filling, unlike on the NHS. This means a dentist can afford to spend much more time on each treatment, if ygm.
Very few dentists have a fixed salary. They get paid based on the work they do.
Hope this answers as to why it may seem Dentists are rushing while providing NHS service.
@DrTSR (who is actually an NHS and Private Dentist) can affirm or correct my answer 👌🏽.