I don't think the issue of you being on time and you taxiing your friend around, are separate issues, I think they are connected enough that they are the same issue. When you are "on time" to the doctor's office, you bear the burden of arriving on time. This doesn't mean you are magically able to predict with 100% certainty how long your trip will take. Rather, it means that you plan in a window of uncertainty of extra time that YOU will spend to ensure the doctor's time is not wasted. For example, if your trip to the doctor's office is about 30 minutes long, you might plan to leave your house 40 minutes ahead of the appointment in case you run into traffic. If there is no traffic, then you have "wasted" 10 minutes, but it's 10 minutes of your own time. It's 10 minutes you will likely spend sitting in a chair in the doctor's office doing nothing productive.
Note that the doctor is not expected to do the same for you. Often, one might arrive at the doctor's office and spend yet a further 20 to 30 minutes waiting, because the doctor's prior patient needed more time than expected. Again, being "on time" doesn't mean that time is never wasted. It means that the person who is asked to be "on time" is the one who is expected to waste their own time so the OTHER person does not have to waste time. The issue of being "on time" essentially includes a value judgement about whose time is more valuable.
People who expect you to be "on time" to pick up your friend, are not just expecting you to be organized. Rather, they expect you to treat her like the doctor: i.e. like her time is more valuable than your time. They expect you to not only drive the 30 minutes to pick her up, but ALSO to plan in extra time in case there is traffic. They expect you to waste your time, but exempt her from the same expectation.
You said that you wanted to give your friend a window in which you could arrive. This would actually be logical because it would make the wastage of time symmetrical and therefore FAIR. If you agreed to pick her up "between 6 pm and 6:15 pm" then you would leave your house at say 5:30 pm, and it would take you between 30 to 45 minutes to get to her house. You might waste the extra 15 minutes on traffic (well I would say you wasted 45 minutes because you shouldn't be doing all the traveling, but we're ignoring that), and she would ALSO waste 15 minutes because she would be ready by 6 pm but might have to wait around for you until 6:15 pm. So a pickup "window" would make you and her EQUAL in the wastage of time. But you said she "prefers" a specific pickup time and gets upset if you are "late." In other words, she wants to be ready by 6 pm and picked up at 6 pm, so that SHE doesn't have to waste time. For that to happen, YOU would have to leave your house at 5:15 pm to ensure traffic didn't make you late. Presumably, if you arrived early, you'd have to "waste time" by sitting outside her house scrolling your phone until 6 pm. She expects you to waste time so that she does not have to waste time.
In other words, your friend expecting you to do all the driving (i.e. you are already wasting an extra hour compared to her), AND expecting to be picked up at a certain time (rather than a window of time) are BOTH symptoms of the fact that she is treating herself (and her time) as more important than you and your time.