I agree that contraception probably played a big part in diminishing the commitment previously attached to sex. And yes, the larger "supply" pool and longer biological clock gives men an advantage in the marriage market.
But I think there's another component to the equation that the video didn't acknowledge. Yes, women do have to compete harder for a smaller selection of men. But I think some women use a different tactic and poach men from the "other" market - men who are not actually interested in marriage. They do this by bait-and-switch, claiming to be in the "fun" market (or whatever you call it) and then end up pregnant. After all, women have the upper hand on birth control, since they have many more options than men, including less visible and less permanent methods. So trickery is actually very difficult to avoid in the "fun" market. And as long as a man has financial resources, and especially a social image to protect, she may very well be able to procure marriage from a pregnancy, or at least many of the same benefits as marriage (financial support, a co-parent, and of course motherhood).
I can prove that women do actually do this on a large scale. Hormonal birth control has a 92-99% rate of effectiveness. Barrier methods are 85-98% effective. (These figures are from Planned Parenthood, ranging from typical use to perfect use.) In the United States as of 2012, 40.7% of all births were to unwed mothers (according to the CDC). In a worst case scenario, if all unwed Americans were using birth control (even imperfectly), we would expect 15% of births to be to unwed mothers. So what accounts for the extra 25% of unplanned births that people so often blame on birth control failure?
I think it's pretty likely trickery. I doubt that most men in the "fun" market would risk pregnancy by knowingly forgoing birth control, and I doubt many women intended to be fully autonomous single mothers. So yes, some women might "settle" for less marriageable men, but I think many women (like 25%) are settling for men who don't even want marriage. And I think if technology catches up for men, and they gain access to a discrete, temporary birth control method like women have, that birth rates will drop drastically.