I'm an academic and have, professionally, interacted with a range of medical professionals. I think you obviously need at least average level intelligence - you've got to be able to process information (often quite quickly), articulate ideas, process numerical information etc. But, you can get quite far with a very strong work ethic and dedication to your discipline. Equally though - there isn't just one type of intelligence - you can be an absolute whizz with numbers but find writing a persuasive argument incredibly difficult or vice versa.
I think the issue of privilege is complex - there are so many ways in which someone can have privilege - it could be access to different educational pathways, more opportunities to meet the right people and have the right connections, or more opportunities to build a varied personal statement/CV. Undeniably, your family having money can access many of these things - whether it be through tutors, private schools, parental knowledge/connections, alumni networks, not having to have paid employment to pay the bills which allows for more unpaid/volunteer work etc.
Myself, I come from a working class background and have seen these privileges play out for others. But, equally, I have to acknowledge that I was afforded some myself. I attended a state grammar school and this meant the focus was on academic rigor. I do not think I would have attended university if I had gone to a different school like my siblings - no-one in my family before me had - am I more intelligent than my older siblings? Maybe, maybe not - but the one thing that was different was that I was given an opportunity that they weren't due to circumstance.
I have witnessed how these privileges can 'smooth the path' for someone who is not quite as intelligent to go as far/further than someone who is more intelligent but less privileged. For example, if you've got money then you can self-fund a PhD and as long as you've got a 2:1 degree then you'll find someone who'll agree to supervise you, but if you haven't got money then you need to get a scholarship which is competitive and only the brightest and/or best ideas get funded. Another example - once you've got your PhD you are often reliant on short-term contracts for quite some time before you get a permanent post. These can involve moving a lot and periods without work - much easier to do if you know you've got money behind you and I've seen some very bright minds leave academia because they couldn't afford to follow this path or because they hadn't been able to do lots of 'extra' research assistant work during their PhD to get the publications needed for their CV (because they were working weekends to support themselves instead). A third example - some applied careers associated to my discipline require postgraduate training that require extensive work experience to get onto the training. Getting paid work experience is incredibly competitive, most have to get it through volunteering, which is so much easier if you've got money behind you. Also, you may well have the connections that means you're the lucky one who gets the paid work even though your family could afford to support you to do the volunteer role! Essentially, being clever/intelligent isn't always enough.