There are scenarios where I've provided proof on the basis that the person I'm dealing with isn't convinced - that does happen, especially when the masking is strong that day, and it sucks when it's necessary; it's very difficult to convince someone that yes, I look totally normal, but we're entering a context where I know it could go wrong.
One was a police incident; the transition from "masking enough to appear neurotypical" to "can't mask this" looks to a police officer very much like the shift from "I'm fooling them!" to "downright suspicious". I have enough awareness of myself and confidence in my diagnosis to be completely up-front about it; it took some time to get there, however, and get past the ego-bruising fact that my mask isn't absolute and perfect. I keep my diagnosis report on my phone for exactly this kind of circumstance; it's pretty well-known that you can't rely on the police to treat autistic folk with dignity without them being forced to.
Another was my employer - a couple of us in the same team both had diagnoses, but the other one refused to disclose it to anyone but me. He ended up being fired for, essentially, autistic behaviour and he preferred to lose his job and get paid for his notice period than tell them (that put me in shitty position, since I was nominally his boss, but his disclosure to me was strictly personal and he explicitly told me it wasn't to be used in a professional context). When I disclosed my diagnosis to them, I could see exactly why - their attitude was awful - so I simply sent them the diagnosis report along with a summary of the relevant parts of the Equality Act. I only lasted a year in that job, mainly because it was a very hostile environment, but still...I had a concrete example of what the environment would've looked like had I not been able to smack them over the head with it.
Which is all to say...yes, you're right that proof isn't always required in many of those situations, but having the option of producing it can be very useful and can actually avoid situations which can deteriorate fast.
It's very true what another poster said, that his primary concern is the social implications - and that's not necessarily wrong. If he is autistic, then his life up to this point has been 90% about blending in, and making sure nobody sees the differences he can see from the inside. It's very, very difficult to get out of that mindset and forgive yourself for all the things about you that you think are "wrong" instead of "different". Hell, I went through this in my 40s and without the raging hormones of the teen years, and it still took me two years to be convinced that I should seek assessment. I'm absolutely glad that I did, though; it's changed my life, and not just in terms of the accommodations I can get if I feel I need them.
Just as an idea, maybe it might help him to actually speak to folk with a formal diagnosis, to see how they feel they've seen the benefits? That could be a bit less adversarial than a parent telling him he should do it...there are probably local charities who have group meetings that might be appropriate for that.