Well, considering there is an abundant supply of free labor, they have every right to send you packing if you aren't responsible (which includes being late). And did you read other posts? It doesn't matter if you are supplying free labor or novel ideas, people think you are irresponsible if you are late.
And the attitude that volunteer positions should not be taken as seriously or treated the same as paid position is why there are so many bitter PIs and volunteer coordinators. They rely on you, and you owe them respect if you commit to something. You are getting a lot out of it in terms of applying to med school, so you should take it seriously.
What does this have to do with anything? Happy and rested innovators could arrive on time.
I think we are disagreeing so much because most of us detractors are speaking more generally than just research.
Yes, in the research field, there is room for flexibility. And most PIs don't care about timing as long as you get results. In the OPs case, however, the PI and OP worked together, so the PI had to wait.
In the real world, pretty much every job focuses on a clock. And as a result, you need to learn to be on time. No excuses. If not, your co-workers, bosses, and patients are all going to notice, and it will reflect very poorly upon you, even if you are a brilliant innovator. You may think that is dumb, but welcome to real life.