Adam,
You are conveniently neglecting to mention Gatwick, Stansted, Luton, London City, Farnborough, Southampton and others.
Heathrow on its own would not be a problem, however there are numerous other very busy airports in close proximity. It is this fact that makes the airspace so complex and, dare I say it, close to saturation at times (though it still has some way to go to get to 2007 levels thanks to the ongoing economic downturn).
There are no doubt much better ways to use the airspace, there is a huge ongoing project at the moment looking into how to do so. The interaction between all the the major airports is what complicates matters.
Until such changes take place, then I personally think that a 3rd runway at Heathrow is smoke and mirrors. It certainly won't increase capacity to the magnitude that some folks say (200,000 plus extra movements per year!).
Of course the OP gave a radical solution - shut the other airports. Not going to happen... neither is the idea of using Gatwick for southbound departures, Stansted/Luton for northbound and Heathrow solely for east and westbound... but if you did that, it would cut out a hell of a lot of crossing SIDS and therefore complexity!
LAX and the surrounding airspace is a totally different proposition to EGLL... the LTMA needs to be updated, and it will be, however as well as coming up with the clever ideas, NATS then has to get it past public consultation, not an easy task.