I think most people who say "no" will have to change their minds as they get older, if they stay single or find themselves single again. As you get older and are dating, the pool is not only smaller, but most people in the pool have been in relationships before and many of them do have kids.
I think it's really easy when you are in your teens or 20s to say, "No way, nobody with kids." But, I think as you get into your 30s and beyond, unless you only date much younger people, it gets harder and harder to find a partner with no kids. Not impossible, only harder.
I have no plans on dating anytime soon, I'm in a long term relationship (marriage) but, I like older guys and would expect only a small percentage of guys over 40-50 to not have kids. As kids don't bother me, it's not an issue. In fact, I think I guy without kids would probably cause me to think twice. I wouldn't automatically rule him out, but as my kids are a central part of my life, I'd want to know why he chose to not have them, or if it simply didn't happen.
I wouldn't, however, date a man who professed to not like or want kids. How would mine
be treated and would he want me to put them on the back burner for him? Not a situation I'd want to get into. Of course, a childless man would certainly be a better choice than a Deadbeat Dad any day. A man who had kids and wanted no part in their lives would be not get a second date with me. JMO.
Mileage and all that.