Everybody talks about the standoffishness of Japanese society, and in fact it did take me some time to get to know these people. But when I did, they really took me in. And it hasn't been for a long time. Geisha exist to preserve traditional art. I mean, I can't imagine they just want to watch them dance.
Most geisha have patrons, yes, for economic as well as emotional reasons.
The relationships with the patrons tend to be long lasting. In any case, their sex life is very much their own private business. I don't think it's helpful to talk about it as prostitution. Over the years I've received a ton of letters from women who think they do. But that's just as unrealistic as the male fantasy of the geisha.