Haven't read the whole thread yet, but I've read about a hundred comments and there's something that no one else has mentioned yet. Part of this strictness in timekeeping in schools is to teach children a valuable skill: punctuality, which as adults they will need to know.
Do you tell your employer "Well, I'm only a minute late, so you're BVU to pull me up" or do you say "Sorry I'm late, I'll try not to let it happen again?"
What you are currently modeling for your children in a sense that the rules don't apply to you. However arbitrary you may think they are, they are what they are, and they have to apply to everyone or chaos reigns. If you teach your children that "this rule is inconvenient so we will fight the consequences for having broken it", that's what they will learn. They will not thank you if, as adults, this is what they know and their working lives are plagued with written warnings for lateness and belligerence when called out.
Okay, this is an extreme scenario, but as PP have said, there has to be SOME cut off, some line drawn in the sand between acceptable and unacceptable. You were late. The correct thing to do is to apologize for being late, follow the school's procedures, and attempt to do better next time. Then that is what your child learns, as opposed to this "I didn't do anything wrong" nonsense you're currently modeling.