I don't know anything about the technical challenges. However, to me the biggest issue would be that it is too far removed from the population that would use it. Yes, it would have good transport links into London, but for those who don't live in central London it would be a nightmare to get to (why are nearly all rail transport plans in the SE based on the assumption that everyone wants to get in/out of central London?).

Everyone who would use this would be located to the N or W of the airport. At least Heathrow serves a wide population and has a reasonably good transport infrastructure going in all directions.

If this mad idea were to get off the ground, is Boris also going to close Stansted and Gatwick if this were to go ahead? Surely it would make more sense to close at least one of these since they only have one runway (both are considering expansion) and would effectively be competing for the same customers.