All your points are true, but then again it is completely realistic because it happens at Gatwick every day.
There's a maintenance hangar the other side of the runway to the terminals to which aircraft are routinely towed to and from - and Gatwick's runway is far busier than Birmingham's.
Equally it's very common for an aircraft to land in the evening, then once the passengers and crew have disembarked for the aircraft to immediately be towed to a remote stand. Whilst these don't have to cross the runway, they do have to fit in between the constant stream of arrivals.
I'm not saying parking aircraft at the Elmdon site and towing them over would be an ideal situation or even a preferred option, but it is an option. Sure it would require additional investment, but an extra tug team and engineer will be far cheaper in the short term than building entirely new stands (and yes I know the handling company would pay for the former whilst the airport would pay for the latter).
It's not likely to happen as the airport has a plan to reconfigure stands and eventually build new ones by the fire station, so the Elmdon site likely won't even be needed in the near future. The current site however is largely hemmed in on all sides, and once fully utilised will either mean expanding beyond the current airport perimeter or developing the Elmdon site.