Well I suspect if you were block off the area under the props completely, you would have a hard time getting off the ground, but not for the reasons you suspect, i.e. the lack of air pushing down behind the props. The reason is because the props do indeed move air and that air has to go somewhere. So if you block off its path downward, you're much more likely to have it dirty up the flow around the boundary layer of the upper prop surface and thus end up with props in VRS and therefore making very little lift.
Now if you to redirect the air flow below props somewhere other than down and somewhere other than back up into the props, say off to the sides or something like that, it would fly just fine. Why? Because the props are airfoils and airfoils make lift by virtue of air flow around the top and bottom surfaces of the foil. That's where most of your lift comes from. Not from the air pushing down.
Now, would doing something like this lower flight time? Of course it would. But again, not for the reason you likely suspect. It would lower flight time because anything you add to redirect air flow below the prop disks is going to also add weight. And any weight you add is going to eat flight time. There ain't no such thing as a few lunch. And that's why adding retracts will lower flight time. Because adding retracts also adds weight. But the gear itself sticking out in the air flow would have an almost immeasurable effect on lift and/or flight time.