Re. the train analogy.
Yes, trains can be late. You may arrive on the platform after the scheduled departure time, and get lucky, and your train is late too, so you don't miss it.
But if the scheduled departure time is 08.50 and you arrive on the platform at 08.51 to find you have missed the train, it doesn't matter that it is late 9 times out of 10, you missed it because you were late.
Re. school start times - plenty of posters have explained why there are safeguarding issues that are behind the policies on late arrival at school - the times may be different, and the policies may not be identical, but if your school has a policy that anyone arriving after a set time has to sign in via the office, and you and your child arrive after that time, even by a minute, you are late and have to sign in via the office.
When the dses were at primary school, I used to make sure they were in the playground in plenty of time - I didn't just want them to be on time for school, I also wanted them to have time to run around with their friends before the bell went, to burn off some energy so they were ready to go in and settle down in class. The vast majority of parents did the same thing.
We even managed to be on time for school when I overslept one morning - we'd had a power cut the night before, and whilst I had reset the time on my alarm clock, I had forgotten that I needed to reset the alarm too - so I snoozed on happily well past the time I should have got up. I was only woken when ds1 came into my room and put the light on - and I snapped at him to switch it off again, only for him to switch it off and then say 'It's 8.17, mum, shall I switch it back on again?' - we left for school at 8.25am!!
I leapt out of bed and into my clothes - and bless them, all three boys were dressed, breakfasted and ready to go - and we dashed up the road and got to school in good time.