Perhaps I can usefully contribute here.
A very knowledgeable US Government guy who was closely associated with the JSF programme explained the X-32/X-35 issues like this: the basic truth that the Hawker/Bristol Engines team understood in the late 50s/early 60s was that any VSTOL solution had to have the most efficient propulsion system possible in terms of thrust to weight - and with the technology they had available, that meant that using cold gas to generate at least some of the vertical thrust was a given. This is because cold gas is more dense and generates more mass flow, and more thrust. That thinking led to the Pegasus concept, which is essentially a medium sized turbojet delivering hot gas thrust with a very large compressor attached to the front delivering cold gas thrust. This basic choice also helped a lot with managing the flow of hot gases around and underneath the aircraft during landing - Hot Gas Ingestion (HGI) poses a massive risk to any gas turbine powered aircraft close to the ground, as the UK found out when they tested Plenum Chamber Burning (PCB) layouts.
Lockheed's concept stuck to that basic thinking, with a separate front fan generating cold gas thrust. With the X-32, Boeing took a different path.
I have to gently differ with a previous post - the X-32 wasn't, in my view, an updated Harrier. Boeing went instead for a fully hot gas solution, betting on more advanced engine technology to deliver a very hight thrust to weight figure, plus very advanced airframe materials to reduce weight - their delta wing design would also have been mass efficient. Unfortunately, almost all Boeing's risks materialised. They were unable to make the advanced materials work, they struggled with severe HGI issues, and their baseline delta design was simply unable to deliver the performance and control margins needed for cat and trap operations for the C model. Having the engine at the front meant having a very long, and heavy exhaust duct all the way to the rear end. Having been forced to extend the aircraft and fit a tailplane, and then redesign the wing, they basically ran out of thrust. Their final design solution using liquid fuelled rockets firing vertically downwards to achieve a vertical landing was, to my mind, an act of desperation.
At the time, I was a strong supporter of the Boeing design, as I liked its layout and potential simplicity - but as Scotty used to say in Star Trek. 'You canna change the laws of physics'. The key point, at least in my view, is this - designing and building powered lift aircraft that can do STOVL is fiendishly, massively, hugely, HARD. Making a STOVL aircraft that's a capable combat aircraft is ten times harder. I came to appreciate the sheer genius of the original British design team (and their US colleagues) who developed the P1127 and the Harrier - and it's that STOVL 'know how' that got the UK into the JSF programme as the only 'Tier 1' partner. As I've posted before, it was also British genius that was the key to making the F-35B work - I had the privilege to see those people in action in the early days of the F-35 and it was a humbling experience.
Best Regards as ever to all those who made this astonishing aircraft work - they deserve our respect. They have mine.
Engines