However, at least during the Alaska Airlines trial, the device isn't handed out until on board and is collected prior to deplaning. Therefore, one can't use it while waiting at the gate nor on a connecting flight.
This may have been a restriction required by 20th Century Fox to avoid theft of the content and/or it may have been a way to allow for on-board rental without requiring a credit card deposit.
It seems to me that if digEplayer-type devices are specific to the airline, this kind of restriction would remain - they would be only for entertainment while on-board the plane, actually probably only while in flight as airlines prohibit all use of portable electronics during take-off and landing.
The program offered by Inmotion Pictures is the only one of which I am aware that actually offers the kind of flexibility you describe.