My daughter who was my foster daughter at 6 did the same thing. And now when I have my granddaughter who is 6 she also begs to stay home with me. Just like you I melt. Fortunately, when my daughter did this I worked and so it seldom was an available option. But a few times I did stay home with her because I knew she felt very insecure. But my granddaughter doesn't have an obvious reason to feel insecure. But she certainly seeks my attention nearly constantly. Since you're experiencing the same thing I wonder if this is a stage. My daughter certainly did not beg me to let her stay home with me after I retired when she was 13. She was well into being inndependent.
I haven't found anything that works any better than just staying firm and sending them off to school. Both of them have so successfully pushed my buttons that I feel teary as they leave.
I wonder if begging to stay home is another effort to exercise control in their life. When you think about it a 6 yo has very little say in what happens to them. Perhaps giving her rewards would help. For example if she goes to school without a fuss she can choose a special activity to do with you when she gets home.
Even giving her a star or a sticker might help. My 6 yo granddaughter always "forgot" to bring home her homework. I think it was a way for her to have control. She was feeling overwhelmed by first grade. I suggested that my daughter give her a star when she brought it home and that worked from the beginning. Getting the star also guarenteed that she had her mother's attention right away. My daughter usually takes time for herself when they first get home.
Do you know if there might be a reason that she doesn't want to go to school. Could she be feeling overwhelmed? or having difficulty with another child or even the teacher? Perhaps a heart to heart talk would uncover her way of looking at going to school or staying home and then perhaps you could then find a solution based on that information.
And is there anything going on that might make her feel unwanted or unloved. Kids this age are so concrete in their thinking. They may overhear
something and give a different meaning to it than was intended. Sometimes I say something sarcastically in a joking manner and Monet thinks I'm serious. And example: Monet was upset because she kept getting timeouts for talking too much. She is a chatterbos and her mother has said such things as she can never stop talking. I asked Monet why she just didn't stop talking and then she wouldn't get in trouble. She said she couldn't because her mother said so. It took her awhile to understand that just because Mom said that didn't mean it was true. A couple of days later she proudly announced that she had no time outs and this has continued.
Casual conversations over time, carefully worded with indirect statements and/or questions has solved more than one situation. Eventually she tells me something that enables me to figure out what may be going on so that I can try a way of fixing it. Sometimes it takes several tries but eventually I figure it out.
And it may be that the situation is resolved just because by talking with her I validated her feelings which then allowed her to work thru the problem for herself.
I don't know if this is any help. This has been a problem for me and writing this has given me the opportunity to recognize some new insights.
I, too, have turned soft as I advance in age. It's hard for me realize how hard nosed I was in my 20's and 30's. What a waste. I think I needed to discipline more with my daughter but didn't because I knew I was too hard while doing it. I didn't know how to relax. I like the way I work with the kids now, better. It's OK to be soft.
Good Luck