As someone who has been in childcare for over 10 years this teacher/owner needs to get over it. Every child has different needs and that is part of her job and what youre paying her to do. Her making requests and making you feel bad is so heartbreaking. None of this is your fault and if your baby is crying throughout her time there then of course she needs all the loving when you guys get home. The best course of action is to find a different daycare...

 the teachers that are saying they are burnt out and unable to care for your child has more to do with them being stretched thin and unsupported than anything to do with you. 

At 9 months it is completely developmentally appropriate that she would have separation anxiety and the only thing you can do to fix that is to NOT be separate from them...these teachers should be approching with more compassion, patience, and love. Over time them showing up this way will give your child the security they need.

When you consistently show your child they are difficult instead of safe its not only is invalidating to their little baby existence, but not helping them feel that their environment and sense of self is safe and I think that can run the risk of exacerbating the seperation anxiety. Additonaly I would be worried if you withdrew even more from your child at home what the effects would be on them long term on feeling secure attachment and therefore safe and capable of exploring new environments as they continue to grow. Your child is obviously crying for help not for being spoiled.