Ummm, no....I wouldn't let my son stay home just because he wanted to.
I raised two kids as a full-time working, single mother. It wasn't always easy on them, but mommy had to work to make money for us. There were plenty of times my youngest really wanted me to just stay home with him, and that would have been lovely, but I just couldn't do it.
One morning he'd been dawdling around more than usual. He came to me in the kitchen and handed me the thermometer. He said, "Mom, you better take a look. According to this, I am technically really sick. I better stay home".
I took one look at the thermometer and said, "Hmmmm. According to this, you are tecnically DEAD. Here's a bit of advice...don't let the water get so hot before you run it under. Better luck next time".
He was shocked and amazed. He couldn't figure out how I knew what he did. (As if he invented that trick).
It really was funny and I hugged him. I told him I wished we could stay home together, but we both had our jobs to do. I promised him we'd do something special when I got home from work.
Now....I have a friend with a 7 year old daughter and the only time that girl isn't crying is when she's asleep. She cries over EVERYTHING and it's a whiney cry that makes you feel like your ears are going to start bleeding.
She has lots of friends, she's involved in sports, she a good student (except for the cry/whining). The mom gets her feelings hurt because even members of her own family will take the two little kids, but they won't take the 7 year old because they know she's going to cry the whole time.
"Some of my tater tots are broken". "My pancake isn't round". "I want a PURPLE popsicle!" If they say they only have purple popsicles and she has the choice of not having one, then she cries and says they're being mean because they won't let her have one.
Her mom has had her checked a thousand times and there is nothing physically wrong with her. I'm not sure I'd say that she's even an overly emotional child. She clearly just hasn't learned any coping skills for dealing with things like a broken tater tot. It doesn't affect the taste. She's going to chew it up anyway. Before you think that she might be a little OCD about things, she dresses herself and could care less if she has mismatched socks on or her clothes don't match. But God forbid if she wants to wear a certain shirt and it's in the laundry.......MELTDOWN. Instead of just wearing the shirt the next day when it's clean.
Your daughter may be an anxious child. I'd talk to her pediatrician about it.
I think she'd do better with learning some coping skills as opposed to just staying home when she wants to. Mirror what she says so she knows you hear her. "I know you'd like to go back to the trampoline place. You had a really great time. It would be nice to go back, but today is a school day. Your teacher and your friends will miss you if you aren't there. Wash your face, get to school, and we'll talk about it more when I get home. I love you and want you to have a great day."
Little kids want to be where they want to be. They want to do what they want to do. That's totally normal. But, it's also normal for them to understand it's not the end of the world if they don't get everything they want when they want it. It sucks, but it doesn't work that way for adults either.
Talk to the pediatrician. He may be able to give you a referral for someone who can work with your daughter on her coping skills and get to the bottom of her emotions. At her age, they do "play therapy" and can really learn a lot about what's going on with a child. If you're concerned about depression, I think this is the best route to go. She can get evaluated without even realizing what's going on. The therapist can help you with strategies.
Just my opinion.
Best wishes.