The extra £500m on an improved rail spur from Slough to LHR can be added to the £9bn or whatever it was for Crossrail as infrastructure invested with LHR explicitly in mind. Add in a spur from the (ficitonal?) HS2 line, and it looks even more ridiculous to suggest closing LHR at any point.
The logical thing to do would build a new 4 runway airport somewhere north of London, on the HS2 line, between the M1 and and the M40. Plenty of space, good connection, and no birds. Unfortunately, lots of historic village and the Chilterns, so lets all just accept that whilst LHR is no perfect, its the least bad option of all these things and get on with it!