If the fluid fuel were cryogenic, we could argue that a leak that allowed the tanks to equalize with outside air would still leave some
of the stuff sloshing inside the tanks - but unreachable and useless, as there was no pressure and no means of driving it into the power generation or propulsion system.
So we could say that the shuttle was using something like slush deuterium for its main powerplant (say, a fusion generator, or the "ion" powerplant mentioned in "The Menagerie", whatever that is), and this in turn provided power to the usual multiple drive systems in forms X (say, warp plasma) for the space drive and Y (say, electricity) for the liftoff gravitics. Further, phaser batteries would store energy in the form Y as well, and so would the shuttle's batteries.
So the shuttle could go to space by powering its liftoff engine with Y from phaser batteries or onboard batteries. Not like a rocket which accelerates madly in order to reach space before fuel runs out (a necessity with chemical rockets), but like an elevator which slowly works its way up but uses very little energy per kilometer or per second. That's what one would expect from an antigravity drive, really. By dumping a few crewmen and the porta-pot, the antigravity elevator would take the shuttle back to space again, even though the main reactor was out of its fluid fuel and could not power up the actual spaceflight engines.
But if a few cavemen tried to hitch a ride, then although the craft had enough stored energy
to reach space, it would not be able to cope with the temporary excess burden, as there was an absolute cap on power
. So Scotty's answer to Spock that the batteries are fine but won't be able to lift them off could be taken to mean that they can't take off on battery power yet
! Not until a few more phasers have been drained into the batteries.
No fluid fuel would actually be added to the shuttle by the tapping of the phasers, then. Scotty would just get more energy into the batteries, in form Y, and this would eventually be used to hover the shuttle and float it to orbit. And once the craft reached space, Spock would be able to spit out the remaining drops of fluid from the main tanks, simply by opening the valves against the zero pressure and zero gravity of space (and perhaps also nudging the shuttle a little with the gravitics).
That would be consistent with the dialogue, too, and would allow phasers to store their power in non-fluid form - say, in the sarium krellide batteries described in the TNG Tech Manual and retroactively established as having been present in 22nd century phasers as well (ENT "Andorian Incident", "Cogenitor").