CGB,
Perhaps I can help.
A key issue with STOVL aircraft is combining an effective vertical lift system with a militarily effective up and away airframe. There are plenty of examples of the former being achieved at the expense of the latter, and also the 't'other way round.
The Pegasus concept is brilliant and effective. It's one of the only two STOVL concepts to get into service so far (Harrier and V-22). But it has drawbacks. The main one is that it forces the engine to be located at the centre of mass of the aircraft, which, if you have requirement to go supersonic, isn't a great location. In fact, for any fighter/bomber, it's not optimal. The Boeing X-32 illustrated the drawbacks.
Second, the Pegasus concept leads to a very fat engine, which again makes going supersonic a lot harder.
Finally, the Pegasus idea seems to 'top out' at around 25,000 to 28,000 pounds thrust. JSF needed more than that. UK and the US spent years trying to get Plenum Chamber Burning (PCB) to work and eventually had to give it up.
The F-35 shaft driven lift fan concept concept requires a lot of doors and isn't as simple as the Pegasus. But it's delivered a viable supersonic seagoing STOVL fighter bomber with internal weapons bays and reduced signature. Which was the requirement. It also allowed a reasonably common airframe layout to be used to address a number of other requirements, which was what the US DoD decided to go for in the early 90s.
If you'd like to learn more, do a Google on 'Bevilaqua', and you'll find some good videos of Paul Bevilaqua explaining the physics behind the idea.
Hopefully, JF will tell me if I've gone wrong in this brief explanation
Hope this helps
Engines