I would without any hesitation dump a babysitter who put my child in car without a car seat. I wouldn't bother to ask questions or educate her, I would simply fire her and move on.
If the other parent of my child did something like this, I would do everything possible to educate him....and I would trust him to comprehend the danger and mend his ways.
I'm not fighting about it as we don't even know if this scenario is what happened (though it makes sense to me). I find it very reasonable to expect a couple to converse about these child-rearing situations and to end the conversation with reassurances to each other and trust. Parents are perfectly capable of mending their ways and being trusted, and I would assume LH was reassured that RH wouldn't do it anymore. Of course, this is all speculation anyway.
I wouldn't bother giving a sitter a second chance, but I would trust the father of my child once dangers were discussed. RH, it turns out, wasn't trustworthy, but she didn't know that. His actions are 100% his responsibility in this case.
He's not so feeble that he can't understand the dangers and needs mommy-wife to do everything. The guy should've been able to handle the task of taking his own child to daycare without someone mistrusting him.
Since she is the more responsible one, then they could have worked out a system to make sure, such as he calls her every morning from daycare.
They both would want to make sure that would never happen again. IMHO