just seems there is no will to do so for whatever reason.
Sad to say, that is certainly the impression that is given. MAN offers explanations about not wanting to disrupt their own operations, stand availability, or handling agents unable to cope with extra unplanned traffic and the risk of passengers being kept on board diverted a/c for hours, but it's so very different to the 'can do' attitude of 20, 30, and 40 years ago. Too much aversion to risk some might suggest.
LAX is right. Why not let ATC know what limited diverts we could take - even if it's only 4 narrow body or 2 wide body, and if necessary on a strict fuel and go basis? Even then, the airport argue 'but a flight may divert in with the intention of a quick refuel and depart, but they may not get a slot back to LHR or wherever, and the crew may then go out of hours'. The perception often is that they look for reasons for not accepting diversions and it does contrast with the approach of some other airports.
Diversions are not the airport's bread and butter, we know that, and they can be an inconvenience, but so can a lot of things in life.