The school run, core hours etc don't really matter. It's all detail that doesn't change anything. It's added because there's always been a tenancy to look down on people who leave earlier or start later to accommodate family.
Op agreed, with a client that this meeting could be at 9am without checking with a 'key person'. That key person is not available to work. Doesn't matter what the reason is.
She knows there's not set hours in the company. So she should know, you can't just assume someone can be available.
Most people would not change their working hours, because someone had gone and agrees a meeting without discussing it before the invite was sent.
My MD, who is my line manager, wouldn't even do that to me or my team. Especially, if we were key people who needed to be there.
Dialing in from outside your home isn't ideal. More so if you aren't just listening.
Op could have spoken to this key person before hand.
The key person could perhaps join a bit later. Could listen in (not ideal) until they get to their desk.
What would the client or have done if 9am was in this person's working hours, but they already had a weekly meeting that could not be moved at this time. Or doesn't start until 10am?
Its really poor to commit someone else to any meeting at anytime without speaking to them.
I suspect ops annoyance is that she will now have to go back to the client and tell them they need to change it.
It doesn't really matter why this person is not in work at that time. They are not in work. It doesn't matter why their working day is structured as it is.
That doesn't mean they are slacking or expecting other people to pick up extra work. Do people have a go at part time workers for not being in when they want them?