You're too focused on blame and responsibility. When a teacher says, "it's your responsibility," they're saying, "ultimately, in life, you are the one who will have to deal with the outcome. You need to plan for the unexpected." It's not a statement of moral responsibility. It's saying, "one way or another, it's your responsibility to be here."
If a parent is consistently dropping a kid off late, the student needs to reach out to the teacher about that. The teacher can then have a discussion with the parents.
You solve conundrums like this by getting away from the blame mentality. We're talking about K12 students being late for class, not people being indicted for felonies. We don't need to obsess so much over guilt and innocence or whether once absence is the parent's fault or the student's fault.