I get that your clock/watch may be a minute or two out of step with the school's clocks - the same applies if you are going to catch a train - or a plane - but this is why I build time for delays and contingencies into my planning when I am going somewhere and have to be there for a specific time.
It was the same when the boys were at school - I knew how long it took to walk to school, and when the bell went, so I planned to set off each day at a time that meant we had plenty of time for the walk to school, with a bit extra built in, and still got there with enough time for the boys to have a good run-round in the playground before the bell.
This meant that they went in with the fidgets shaken out of them, ready to settle down and start the day - they didn't rush up to the school and arrive late and flustered, and need time to settle before they could get going on the day - time that would come out of class contact time.
I am routinely early for things - if I have an appointment, I look at the journey time, add on time to find parking and walk to wherever I am going, and then add extra time to allow for delays and problems. I then usually add a bit more extra time, just in case - so I get to places early, and sit outside with my book. I do understand that this doesn't work for everyone - I don't usually have back to back appointments, so I can leave earlier if I want to, and it doesn't matter to me if I end up wasting a bit more time waiting - but other people have tighter schedules. I still think they can plan to get to places comfortably on time, rather than just on time, flustered and racing the clock.