I think it may be poor wording, where "single" is used in place of "available." They're not the same thing, but the implication you're reading is that single people are always "looking" as opposed to "just doing their own thing right now." People do actually use "I'm single" to communicate availability, so I don't know that that conflating the two is intended, or directed at you and yours. But I get where you're coming from that it could seem like it's stigmatizing or putting you in a box. That sucks.
I'm also not sure this is really a gendered thing on the whole, though. If someone, male or female, shows all signs of being an incredibly good partner--hot, charming, exciting, rich, whatever--AND it seems like you have a shot with them, I think plenty of people can easily get overexcited. After all, there are great, fulfilling relationships, and then there are "dream relationships," and most people's lives, realistically, still at their best only get the former. So if you're one of the lucky few who gets something seemingly special heading your way, you might want to rush in and grab it. Sayings like this (and many others) are, I think, a collective attempt to remind ourselves to keep our heads. Don't get blinders on. If someone seems superduper great, you're probably not the first person to have thought so. So maybe you should do some more thinking. People make a lot of excuses for rich, charming, attractive people. None of us should presume we're not at risk of doing the same.
It's true that some versions, like the what you quoted, really leverage cynicism and negativity to pull that off--maybe since cynicism is pretty much the opposite of excitability. Maybe there's a legitimate discussion to be had about whether that version is improper or societally damaging, even assuming it's (more) effective. But I do think that even in those negative cases I don't know that it's "hateful", or that negativity itself is really the goal.
Not saying this is certainly it, but it seems to me to fit how I've encountered this thinking. Maybe others would have different experiences.