If you dismiss people who are asexual - which is an odd choice on your part, as you are one of them - then realistically, your only chance of forming a romantic relationship with a man around your age who doesn't want sex will be with a man who has zero libido for some medical reason or other.
Personally, I think it's unlikely that many - if any - men in that situation are going to be using dating sites unless they are sites that are for, or at least accommodate, people who are openly asexual.
But you obviously won't know unless you try.
Are you seriously saying that the only thing a man could get from a relationship is sex?
Obviously men also want love and affection and companionship - but that doesn't mean they don't want sex as well.
If someone has a libido, they're not going to want a new relationship that doesn't include it. Of course sex isn't the only thing men might want from a relationship, but it's certainly one of the things they're likely to want, and that's completely normal and reasonable of them.
I'm not a man, but if I found myself single and looking for a new relationship at my age (49) I wouldn't even entertain the idea of dating a man who didn't want sex. If my existing partner became unable to have sex, I wouldn't leave him because I love him so much, but that's a love that's been built up over nearly a quarter of a century. I wouldn't sacrifice it for someone new, not in a million years.
I think you would do well to have realistic expectations - and to stop being so weirdly dismissive about the asexual community when you are, apparently, asexual.