Programming is not sports. I'm not a coding athlete. Maybe it is for some people, but not for me. That's not what the job I've held for all this time has ever been about.
Plus you get to a certain age and experience and the thought of "proving myself" to some 25 year old is just... Really?
If you just don't like hand writing and prefer typing, then that's cool with me and you can just do that during the interview (the content of interview is important, not the medium you write with).
If you're mentoring a 25 year old you'll have to prove yourself to them every day.
No, you won't. If they don't want to listen to you, they don't have to. As a mentor it's my job to listen to them and guide them with experience as my context, not joust with them over irrelevant, stupid, gotcha contests. And I'm happy to help them realize that just by not engaging in it.