Its horses for courses. Med school courses vary, and students vary. And medicine itself offered a good variety of careers.
Medical schools will not want to use the same selection criteria, as then they would all be picking the same students. They will also want to select those students that are right for their courses.
The NHS will be concerned that those selected know what they are letting themselves in for, and will want to remain within the NHS long-term. An Oxbridge educated friend says that she was the only one of her friendship group still working as a doctor 10 years after graduation. Some will have gone on to be sucessful in other fields using their knowledge of the health industry: law, investment finance, etc, but this is not what the NHS wants.
So the NHS want to see evidence that that prospective students know what they are letting themselves in for (it not all Gray's Anatomy), how the NHS works, and that they can cope with it through the longer term. Hence the guidance posted by MedSchoolRat. Some of the more hands on courses will presumably then be looking for more. This too is fine. DD did loads of volunteering, essentially because she is quite a hands on person and will probably be that sort of doctor. She much preferred doing practical stuff with people than practice for UKCAT. This limited her choices, but also meant that she is on a course where they are observing from the second week, and where before starting her second year, she has already done a number of HCA shifts, including one an acute ward where, sadly, someone died. She is expected to do a full month of HCA shifts, including nights, before next term starts. She is also required to volunteer in her student City.
Stranger may correct me, but I understand the first year of the course her son took, has a different approach, and would have required different strengths from their 18 year olds. Given Stranger's DS is clearly very bright, my guess would be that the one offer he did not get, was from a medical school that was looking for a different ratio of practical experience to academic.
OP your son needs to do a minimum outlined by Medschoolrat and if he is reluctant he will not be the only one to treat shadowing/volunteering as a box ticmking exercise. The skill required is to be able to talk about it enthusiastically. However he might turn the decision round. Is he happy to work with old people? With children? With the severly disabled? Is he calm in a crisis? Can he cope with bodily fluids? Can he work in a team with people from a great diversity of backgrounds? Is there anything he wants to test? What might help his confidence going into an interview? Is he likely to want to apply for a hand-on course? From memory Lancaster and Bristol were the ones who wanted evidence of sustained volunteering, though this may have changed.
DD spent a summer (post yr 11) working in a care home cleaning, kitchen portering and waitressing, which she enjoyed. (She started as a volunteer, until the other support staff complained on her behalf that she was being excploited and should be paid. Fair enough. They kept adding shifts till one week she did 7 days of double shifts.) She then did a ski season during her gap year working for a family ski company, where she found dealing with family dynamics one of the more difficult parts of the job, something that might affect future career decisions. She helped out with a weekly disabled swimming session at our local leisure centre for a couple of years, which was really rewarding. One-to-one including hoists, helping overcome fear etc. (There are volunteering opportunities beyond care homes - and leisure centres are a good place to ask.) She also did a week at a summer camp for the disabled at the end of year 12, which was tough, but which she again loved. (Happy to PM the name of the organiser - they have volunteer run camps across the UK, plus volunteers run weekly activities in some cities.) Indeed this summer she is repeating the last two activities, not because she needs to tick boxes, but because she wants to.
She also did three shadowing stints, arranged through friends. These were useful to help her understand what doctors actually do, but my assumption is that not everyone has useful friends and so medical schools will be wary of expecting it.
I am sure that some careers in medicine will echo careers elsewhere. Go to a top University, get a first, be accepted by a "Magic Circle" London deanery and in your preferred specialisation. Continue to progress with the aim of applying for a consultancy at a well regarded hospital in your early 30s, and so on. This was not what DD wanted and she chose not to sit BMAT, thus ruling out Oxbridge/UCL/Imperial. She will be fine, Stranger's son will be fine. OP's son should think a bit about what he wants from a medical career. If his dream job is as a GP on Shetland, he should look at St Andrews and read their entry requirements carefully. There is little point volunteering just to tick boxes, but lots of value to be gained from good volunteering experience and knowing what you want from it.