You could offer to go into the before school program and let her show you all the 'stuff' that is there. (Call for 'permission' first, but it would be very bad form for the daycare to refuse.) Then you can show your child all the things she could do. Maybe one morning you could take a fruit snack or bubbles to help your daughter engage the others.
I can tell you that when I was a child, saying that I was bored led to being kept very busy.
There were always toys to put away or a pup to walk in our garden, or laundry to put away or dusting -- all things I could do by age four.
In my classroom, an idle child was given a task. If they found an activity, they were allowed to carry on until the next work period.
Perhaps you could ask the daycare to make sure your child is busy. If she finds something on her own, great. If not, she can tidy a shelf, or put toys away.
As to being late, that is unacceptable. Tell her that if she makes you late, she will owe you the time back. It could be she goes in her room to read before the usual time; it could be chores she has to do; it could be waking up earlier; it could be time away from something fun you do on that same day.
Let her know by using 'choosing language' that she is responsible for the consequences of her actions. (This is your choice. This is his/her/my choice.) 'Choosing language is not used negatively. "Do want eggs or waffles?" "Yes, you chose waffles." This is not an excuse to be angry. It is okay to commiserate. "I am sorry you chose that instead of this." If you are calm and matter-of-fact, you give her no reason to fight. This is simply the way it is. Don't back down once you've decided on an action. Be calm. Don't fight.
Perhaps make a commitment to do a fun activity after school -- it should be something you enjoy doing -- a bike ride, a trip to the playground or a board game. Devote an hour or so to that activity. Remove time if she 'spends' it elsewhere.