I do hear what you're saying, OP, but I think this rhetoric of 'good single mums stay single ' makes the alternative 'only bad single mums date or want sex'. This good vs bad is what puts us back in virgin/ whore territory and we don't want that.
FWIW I've been a single parent two years and not dated at all. My friend has been dating and had a few serious relationships. We are different people, she can't be alone whilst I can tolerate lots of time on my own.
It would be wrong to judge her for wanting to be in a loving relationship, we're not all happy with a book and a gin like I am. I don't think we should judge people needing that intimacy and reciprocal bond that you get from being in a relationship. I love going on holiday on my own with the children, whilst she would just hate every second, it's horses for courses. She's not a worse mum for not wanting to live her life on her own.
In the same way, I hate it when married women or women in relationships comment that 'if they were to split up with their DH they would be single forever' but they don't know that. They don't know that they would be happy living like that, because they haven't experienced it. They are just fed up of cleaning up all the time and the lack of thanks, but they have no idea if they could hack spending every other Friday night, Saturday night, holiday, some Christmas's on their own (in a 50/50 custody split scenario).
I'm sure many of them would change their minds.
Is it better for the children? Well that again depends on the parents. If mum is depressed being single, and doesn't feel able to do X,Y,Z on her own, then no. If mum thrives, has lots of friends, enjoys her job, gets out and makes memories with her children then probably yes.