the meaning of Er hat sie gestern zum ersten Mal sitzen gelassen.

I asked my german friend about this and she said yes it is an idiom. But the literal translation would be better as he left her sitting for the first time yesterday. 

She said she might use the idiom when someone is stood up, or someone didn't turn up for an arrangement. It could even be used when someone is dumped.

So I have the impression that it has a connotation of separation.
My understanding of to let someone down is that it has a connotation of disappointment and can be used in the above circumstances but also more widely .