BMA is one of the weakest unions in terms of fighting for the rights of its members but it has an undeservedly fierce reputation as being a strong union.
The reasons doctors voted to not increase medical schools and medical school places were complex and boiled down to obvious lack of funding and totally inadequate planning. The proposals at the time we're to increase medical school places in existing medical schools but without a corresponding increase in funding, facilities, lecturers, or clinical tutors. Essentially cramming more people into the same class with no extra resources. Unfortunately, medical training is extremely resource intensive. You simply can't have 20 medical students filing past a sick patient, all of them examining the same person being supervised by the same person. You need more people teaching, and teaching students takes time away from other clinical activities so hospitals don't like doctors doing much teaching unless there are enough resources.
Proposals for the new medical schools were put forward to lower the teaching, assessment, examination and exit standards for the newer medical schools and would create a 2-tier medical graduate system.
There were also no plans to increase junior doctor posts for these doctors to be employed after graduation, no increase in training posts so that they could receive the necessary post graduate training they need to become GPs or specialists (a GP still needs 5-6 years of post grad training after 5-6 years at medical school, some hospital specialties need 6-7, others need 8-12 years additional training), no plans to increase consultant posts so that any newly trained specialist could be employed, and no funding for increasing GP numbers.
These were just some of the more easily understood issues with the proposals at the time, there were many others including changes to the training scheme application system, qualifications awarded, specialist examinations, loads more. The 2008 vote wasn't just about medical school places or new medical schools, but entire medical careers, specialist medical training, working patterns and hours and other linked issues. Many of those proposals were implemented anyway and wrecked tens of thousands of careers with huge numbers of UK doctors emigrating to Australia and Canada, which meant we had to rely even more heavily on doctors from abroad.
It's far too easy to say just train more doctors, or condemn a vote from 14 years ago for opposing proposals to increase numbers without understanding the underlying issues. It takes a huge amount of long term planning and investment that's way beyond the horizon of any government - 12 to 18 years at a minimum - to be able to get us out of the hole successive governments of all hues have dug into.
DOI: I'm a doctor and had many friends affected by the "medical career reforms" around 2008, the MMC/MTAS fiasco, which devastated the training and careers of a generation of doctors.