Lol--OP I have 6 kids and have travelled with a 6 month old and 18 month old by myself.
We never had an issue with a child of any age refusing to carry their backpack.
You seem very committed to your roller board method so maybe you should buy MCE so you are guaranteed overhead space?
And the root cause is you arrived late to the gate which meant the overhead space was full. If you had arrived at the gate an hour earlier none of this would have happened.
Like others--I don't know what exactly AA's policy is for gate checking bags all the way through vs. to the connection.
Your roller board is NOT your personal item. A personal item is one that fits under the seat.
The rude gate agent should be brought to their attention.