I asked a similar question before we went.
In the end we took our printed confirmations with us for peace of mind, but dodn't need them. I asked at VA check-in and they said their systems currently don't have any ESTA info, but in future they might.
The immigration officer at MIA checked our green forms and tapped around on his computer, but didn't mention ESTA.
My feeling is theyre parallel-running the ESTA with the green forms until they're confident it all works. Then I assume it'll be a bit like APIS where it's all linked to your booking.