I met my DH as a single sahm. I also have a disabled DD and I have disabilities myself. It's never really bothered my DH, and he certainly has a lot of respect for me and we have a fully equal marriage with full access to finances. DH tended to pay for everything when we were dating, so it didn't affect what activities we could do together. He has a professional career and earns a good salary (6 figures), enough for me to be a sahm in London. We met through a sporting hobby. I never went out with the specific purpose of meeting a partner, let alone a professional one - I wanted to improve my fitness and develop sports skills and meet friends in general.
We were attracted to each other because of our looks, personality and sense of humour, and things like my family or financial circumstances didn't really come into it. I am lucky to live close to my family so childcare for evening dates were never a problem, and we were able to go out to do interesting things like going to gigs or theatre and not just sit at home watching Netflix just because I couldn't get a sitter.
I claimed benefits to support me and DD when I was single, and I lost my means-tested benefits, including child benefit, when DH and I got married, and he had no issues with combining finances so everything was family money. I still receive carers allowance, DLA and PIP. DD's dad isn't involved in her life and doesn't pay maintenance, so DH regards us all (with the dc we have together) as one family unit and doesn't treat her differently.
OP the vast majority of single mums do end up meeting new partners in time, the average length of time for a single mum to stay single is only five years. So there is a good chance that you'll meet someone, but I agree that I've been lucky to have childcare available because physically getting out to meet someone is half the battle.