It's probably because I'm a dinosaur, I realize that, but it just feels weird to me when somebody I've never exchanged a single message with, who doesn't speak a language I wish to learn, and who isn't even learning one of the languages I'm learning, asks me to be his/her friend - and I never hear from them again. That's why I've disabled the button.

Plus, I just don't see what it's for. There are people on italki I've been talking to regularly for years. And none of them has ever even asked to be my "friend" here - there's no need for that, because we've actually become friends in real life.

At times, people are overwhelmed on this site.  Maybe that person you were contacting gets blasted with many friend requests from people who aren't serious about being exchange partners.

Please don't hold it against them.  Like many have said before me, a solid introduction is usually preferred before going in for the friend request.  

After a while of being here, I stopped routinely accepting friend requests (and turned off the button to discourage people from sending me requests,) because I was worried that for some people, accepting a friend request may be perceived as making a commitment to help them regularly.  At first I just accepted friend requests, but after seeing people post here their disappointment with ¨friends¨ who are not really commited, I decided I preferred not to disappoint anyone.

I leave it possible to for someone to send me a message.  Now when someone has corresponded with me through private messages,  I very occasionally suggest we become friends to make it easier to contact me if I feel there are benefits to both persons.