The real problem is that the incentives are wrong for the sites. If they only got paid when there was a match that resulted in at least 3 months of dating or friendship then you can bet they'd actually get that kind of result.
Instead they actually get paid by letting men message women irrespective of how appropriate the match is(1), and by getting page views by suckering in people to believe more "connectivity" is available than really is there.
If there was a way to reliably match people algorithmically, we'd have seen it implemented by now. Sure, people would leave the site faster, but it would also provide an EXTREMELY in-demand service, allowing such a site to charge massive one-time fees.
If I sold a bona fide 95+% chance of finding one's perfect mate for say $1000, then I would have a LOT of happy buyers. Problem is, it can't be done. Or at least it would require some kind of human-level AI.
You wouldn't have a lot of happy buyers - you'd have far fewer. You are assuming that everyone on a dating service has a good match on that same dating service, which isn't the case. From the business side you have a choice, get a small amount of money from most participants or get a large amount from very few. Even if the totals are the same, the latter is far more risky.
There are quite a few human run expensive dating agencies. It is what rich people use. This one for example costs about $100k to sign up.
http://www.orlythematchmaker.com/
An article from a few years ago titled "Professionals pay matchmakers to be headhunters for the heart" about expensive human run dating agencies:
Instead they actually get paid by letting men message women irrespective of how appropriate the match is(1), and by getting page views by suckering in people to believe more "connectivity" is available than really is there.
(1) Explained in an OKCupid blog post they pulled but a copy is at http://interestingreads.posterous.com/why-you-should-never-p...