Absolutely a "deal breaker," in my opinion. If they are a licensed daycare, they cannot transport children without your written permission, and, say it with me, CHILD SEATS! If you don't want to leave, but want to draw their attention to it, submit an anynomous complaint to the licensing board and they will do a random drop-in. The providers are supposed to get them on occaision anyhow, so they'd never know the difference.
I will say one thing... You are her mom, go with your instinct. The first home daycare we put our son in was recommended by a friend. Her play area was downstairs, but she had the kids naps in pack-n-plays upstairs in bedrooms. Our son was only 6 weeks old and I was uncomfortable with the arrangement and asked if we could bring our own pack-n-play for her to keep downstairs near her. She agreed and that is where she was to keep him. For various reasons, we left the daycare. We found out a few months later, via some other friends and then the news, the baby that took our son's place died in her care because she put the baby on the bed downstairs to sleep. The baby fell between the wall and the bed and suffocated. She had been there they estimate for over 40 minutes, and no one had checked on her. I thank the Lord every day that I followed my instinct or she'd have put our baby on that bed. Seems it was too inconvenient for her walk upstairs to put the baby in a pack-n-play. Just as it seems too inconvenient for your providers to use child seats.
We ended up trying another home daycare, then resorted to a center which I swore I would NEVER do. I will tell you, I wish I would have started out with a center. We LOVE it and so do our boys. I will not go back to home daycare. The centers have even stricter rules to follow, have their own play areas, and our does not transport children. They have "field trips" come to them. They've had the fire department, police, race cars, clowns, etc, all come to them.
PLEASE, trust you instincts, not other peoples opinions and judgements. And most of all, please don't let them transport her without a car seat. I try to avoid anyone transporting our kids. (Oh, and I am with you on the stair gates also...)
Good luck to you, but don't feel "over protective."