Never had anyone beg for it. That does seem a bit odd, but then again there are good people out there whose simply had experiences (or lack thereof) that lead to some, in my opinion, awkwardness in interactions.
That said, I do feel that true friendship isn't something that comes from putting a title on it. Going "wanna be friends? okay" doesn't actually establish a friendship... the actual interactions do.
Whether or not I'd give them a chance is something I'd have to make a gut feeling decision on. Without knowing anything else, chances are I probably wouldn't. Nothing inherently negative against them per se, I just think that if that is their "social language" (for lack of a better term) then we're very likely far too different to mesh well together.
And my social batteries are very low too. I already barely have enough for the very few friends and family I do have in my life, so I'm picky, for my own health and the health of current relations. Maybe they're a good person, I'm sure there are folks that would be a great friendship match for them... but most likely not me.
If I did give them a chance though and it just doesn't work out... well, I do find that tough to deal with. If we don't gel that doesn't mean I think them a bad person, that they are awful or undeserving or not worth anything. Just different pieces of the jigsaw puzzle of life. Doesn't mean there is something wrong with them, just not the right fit.
But at least in places like this there are a lot of people who are fragile and/or will take it personally if you do tell them "this isn't working out". On one hand I'd want to say that to them, on the other I don't want to them to end up feeling discouraged or afraid to try to build relations with other people later. At the end of the day however I solve it I do have to look out for myself.
The way I see it you don't become a friend by asking. Calling someone a friend is more a way of expressing a feeling and explaining a connection over time.
