Depends on how old it is, I definitely don't recommend it with an adult or any wild one, but a kitten at a zoo, you could definitely survive. I have even seen the workers pet and hug tiger cubs and it was part of the work.
Well even on not being rare, lynxes are stalking predators. Given what noisy clumsy travels us humans are and their keen senses, one is lucky to see a lynx. Since firstly they are always stalking or hiding just naturally and specially so hiding upon most likely spotting human way before human spots them. One could go right by one and not notice it. We aren't on their menu given our size and not being normally encountered prey species. Also as stalkers unless it is something like a mother lynx protecting its young, it won't make itself known. Far rather hides and let's you pass without encounter. Since one less risk of the lynx getting injured in fight, if it can't just hide away and go unnoticed.
Though on top of that some species of lynx are very endangered.
I know what you mean. Been touring the mountainside for years and i only managed to see one once, it was condescendingly judging me from a cliff above as i relieved myself in the bushes.