Phlebotomists are employed to do a specific job and don't need qualifications of any kind, the pay is around £9.50 an hour, with on the job training.
The phlebotomists in my nhs trust are all based at the large university hospital but travel to different surgeries on a rota to take blood locally.
Even so, the current waiting time for an appointment is around 10 days.
The nurses at my GP practice give the routine injections - flu, shingles, pneumonia and now Covid boosters. If they had to take blood samples as well the waiting times for the injections would increase.
I think the system works well, overall.