Be on time... I’ve got other things to do!

You are not being unreasonable. In your position, I'd be irritated unless I found out there was a proper reason for chopping and changing.

No you are not BU.
I see no reason short of losing one’s phone why it is not the norm to let someone know if you are unavoidably going to be late, whether it is friend or family, or to ring and cancel if you no longer need an appointment. Otherwise you are saying, My time is my own and I don’t care about yours.
Especially bad in the NHS. Quite a few No Shows at CAB too.

Sometimes our children don't realise we still have busy lives even though we are retired. Maybe make it clear in future? I hope it eventually worked out.

Ah yes, I know what you mean. It's the "waiting in for the plumber" scenario, only for you it's family.

If I have a situation like this, as I have, frequently, I leave a note on the door, in an envelope with the visitor's name on it.
"I'm so sorry, I thought you were coming at ........ and I've been unavoidable called away. Please ring me (or I may say "I'll ring you") '
Then next time I have contact with that person, I advise them to PLEASE let me know if, in future, they've been held up later than the arranged time.
All this can be done with kindness and humour.

But isn't it annoying until you've got things clarified?