Thames Estuary did not proceed for the simple reason that not enough pax live that side of London and all the ground routes lead neatly into EGLL, due to history.
Many pax come from the central and southern part of the UK, the Shires of: Oxford, Warwick, Buckingham, Hertford, Berk and Hamp, as well as Surrey and Sussex. Whilst many Londoners could cheerfully go East or West to get to the main airport, it is all the rest that cannot. If Thames Estuary was built, you will still need a field somewhere to the west of the capital. Airlines would not run flights from two airports and the problem would not be solved.
Further, property (both domestic and business) has been built and priced around EGLL, West London and the M4 corridor for 50 years. If you cut the traffic of the airport in half, you would remove profits and investment from millions of people and companies. No politician is going to do that.
LCY took care of much of the European commuter problem for the City and has done well for many locals in Essex and Kent too.
So ... expand EGLL or leave it the same? Personally, I agree with the view that they have missed the boat (pun intended) and the peak of air travel will be over. Keeping an artificial cap on EGLL expansion makes it more profitable than a larger site operating at less than capacity. BAA will find that they might encounter the law of diminishing returns. The 3rd might be needed but it is not required.