I have the musical ability and quality of a blocked nostril. I would choose the violin. It seems to have so many moods. It can be happy or very sad. It seems as happy making glorious classical sounds that can make even the most stony hearted weep or, at a fireside party.
If it had to be modern, I find myself drawn to the bass guitar.
Funnily enough, these two are what I play... though to be honest I haven't played the violin in quite a while, it's just too noisy in a small house. It takes an awful lot of work to get to a listenable standard on the violin, and a fair bit to maintain it - thankfully I learned as I was growing up and had my parents to keep me from quitting when it seemed too much, there's no way on earth I'd ever be able to learn it to the same level at this age.
I love playing the bass though - not that I'm good, mind, (I only started two years ago after years of playing guitar, also self-taught) - but good enough to be able to play what I want. I only play for myself, almost always in the evenings through earphones to avoid disturbing the rest of the household but it's a great way to unwind. I have a very nice Rickenbacker guitar which has barely seen the light of day since I got the bass - I think four strings is my ideal!
I fully agree that although some will naturally have more potential than others, the vast majority of people who really wish they could play an instrument could do so if they were willing to put the effort in for enough time.