Recruitment Consultants get paid by employers so our value to them is a 'placement' fee. It's not the end of the world if one agency doesn't think their clients (either the ones they have or ones they want to have) aren't suitable. There are others.
For what it's worth, I think you didn't do yourself justice when you met with them (well, that's the way it comes across in your post).
Firstly, you said you needed to do the school run, which implies part time working 9:30-3pm, rather than a standard 9-5pm, but picking up your children either side.
Secondly, if you work 35 hours, that is full time in lots of organisations.
Thirdly, you said you need needed to be with 1/2 hour of home. All you had to do was just say that you were looking for roles in xxx area or that you were only willing to commute for 1/2 hour.
I've been placed in a couple of roles through agency's and my circumstances are similar to yours. When working with agency's, ive just said 'I'm looking for xxx work, in xxx location and xxx salary. If something comes up, let me know'. I am open about my family life, but I certainly haven't apologised for it.
The employers I have worked for have all understood that people have lives outside of work and it hadn't been an issue.
Whilst agencies can find you work, you really should apply your net wider - apply directly to company's, use your networks, get in LinkedIn.