Not only is there nowhere suitable in SE England for a large new airport, but the past history of the few major ones that have been built (Montreal Mirabel, Tokyo Narita, Osaka, etc) is that airlines are very resistive to use the places, particularly for short-haul, and knowing only too well their commercial markets (something that politicians ignore) will do anything to avoid using the place. And the passengers likewise. If you end up with a split site that kills all connections, as Montreal found out. In fact Mirabel was the making of Toronto as Canada's main gateway; before Mirabel Montreal had been, and a significant rationale for Montreal's commercial decline and Toronto's ascendancy was Mirabel (now closed).
Look at what happened in Dallas when DFW was opened, all the main carriers moved across from the old Love Field and little Southwest quickly built up an enormous business based on those who preferred not to go there. It now handles more flights that it ever used to before DFW opened. Meanwhile Prestwick markets itself as "Glasgow Prestwick". Hmmm, that doesn't seem to work.
Yes you could legally enforce the closure of Heathrow, but the political fallout from employees, property owners, frequent travellers and businesses who have developed all around it would make the 3rd runway issues trivia in comparison.
Such proposals commonly include a "high speed rail link" to one place in the old centre of the metropolitan area, generally which has no car parking, disregarding that very few originating passengers come from there or have any desire to go into the centre to come out again, and a notably small percentage of inbound pax as well.
The commercial advantage would lie with any carrier who managed to stay at Heathrow. So they all would want to.