Good suggestions below. Sometimes kids hold it together when they are in public, and let loose on parents.
But she may also be doing this because she CAN. I totally understand that you don't have time to fool around in the morning because you have to get to work. But coming home from the program at school when she wouldn't get out of the car? If you have a garage and it's safe, I'd have left her in there and I'd have told her to come inside when she was ready. I'd have taken myself - her audience - away from the situation.
You say there's no talking to her when she's upset - I agree. Don't do it. The thing to do with a tantrum is ignore it. Now, you can't allow her to have a meltdown in certain places in public (church, a wedding, a funeral, the office), but you can remove her from the situation with as little fanfare as possible and absolutely not engage with her. I've left my kid in his carseat or booster, stood outside the car with a book or a crossword puzzle, within sight so he knew I was not abandoning him but far enough that he knew I wasn't listening to him either.
I agree with a code word (such as "Are you arguing with me?") that informs her that consequences are imminent. And then follow through. Losing privileges next weekend for an infraction today won't work at this age. But losing out on a story before bed (or a half hour of TV or whatever her pleasures are) because of a tantrum she threw an hour before DOES work.
She may also like having you carry her and fasten her seatbelt. She has some control over you that way. So make it unrewarding enough by ignoring her completely, and just do what is necessary to keep her safe. So, if she won't put her coat on in the morning, take her across the street with to B&A care with no coat. She'll live. She'll be cold but she will survive just fine.
We had a parents' workshop with a facilitator on just this topic. The expert advice was to adopt the attitude of "How unfortunate for you, my child." The child has to learn that decisions she makes can backfire. Example: Child refuses to put on gloves to go to recess even when teacher instructs. Rather than get into a battle, the teacher tells the child she will not be able to come back in to get gloves if she changes her mind. Of course, within 5 minutes, the child has cold hands. Teacher on recess duty says, "How unfortunate for you. MY hands are nice and warm because I put on my gloves." So child has to stay on playground with hands in pockets, not playing. Teacher does not give the child her gloves, does not send anyone in to get the gloves, does not engage in further conversation with child. Next day, child wears gloves.
Same goes for giving in to a tantrum in the store. I've checked out of a grocery store with only half of what's on my list, because my kid had a meltdown over not getting a candy bar. I don't get the candy bar, but I also don't buy the things my kid enjoyed that weren't essential. "How unfortunate for you. You had a tantrum in the store so I couldn't allow everyone else to listen to your screaming while I shopped for juice boxes and your favorite cereal." And later, "How unfortunate for you that I don't have time for X because I now have to go back to the store for what DAD AND I needed and I didn't have time to buy due to your tantrum." No endless lecture, just simple facts.
So, your child can go to daycare with or without a coat, but she can't opt to stand there and fuss, thereby controlling YOUR day. She can choose to fuss and cry with the providers (they're used to it) but she can't make you late to work. She can choose to sit in the garage bored to tears, but she can't make you stay there and carry her in the house.
It's okay for kids to be unhappy, angry, frustrated, and disappointed sometimes. They have to learn to calm down and get over it. We can't always make them happy, and experts think we shouldn't all the time.
And believe me, the other kids do similar things with their own parents. You just aren't seeing it.