I think what you are leaving out of your analysis is that, in this scenario, there is no compulsion to help the beggars. Thus, if everyone actually is helping they are doing for their own reasons: that it is good, that it is right, that they want to help. I find it hard to imagine that a significant number doing so would then take advantage of the same system in the manner that you describe.
"But", you say "it is ridiculous to even assume that everyone is going to willfully give as such in the first place." This much is obvious, so trying to comment on the above utopian ideal has its limits. I am inclined to believe, however, that if everyone who actually would be willing to help beggars of their own accord to make things better did help that it is likely that this would only be done in such numbers that the beggars could be helped but not such that it would become such a lucrative market as posters have described.