I believe the paradox you point out is taking it to the extreme. This is the parent poster's sentiment: You don't care deeply about a rejection outcome, but you

I believe the paradox you point out is taking it to the extreme. This is the parent poster's sentiment: You don't care deeply about a rejection outcome, but you care enough about a positive outcome to give it a shot and see what happens. If it happens to be the case that the receiving party rejects you, you should not care and move on easily. I think the benefit of making a good friend far outweighs the cost of rejection, so much so that we are willing to give it a few hundred shots. For some people, the chances are so low that the chances range in thousands, not hundreds, and then they start giving up.