I had a quick look at the Pico project but got confused, if I want a basic set of buttons (flippers, magna, navigation), a plunger and accelerometer, is that all handled with a single Pico board?
Yes, except that the Pico doesn't have an on-board accelerometer, so you'll need an external one. The easiest one to use at the moment is probably LIS3DH, since you can buy a nice little breakout board for that from Adafruit or Sparkfun for about $5. It's pretty easy to hook up with the breakout board, but it's not as neatly self-contained as the KL25Z, obviously, so I wouldn't recommend it unless you're comfortable with a little DIY wiring and configuration work. The downside of "highly configurable" is that you have to configure it. The plunger connection will also require custom wiring.
As for buttons, the Pico has 26 exposed GPIO ports. You'll need two of those for the accelerometer, and if you're going with a potentiometer-based plunger, you'll need one more GPIO for that. If you don't plan on any feedback devices, that will leave 23 GPIO ports that you can use for button inputs. That's enough for a pretty well-equipped pin cab. If you wanted to add even more buttons, or if you plan on also adding feedback devices, things get more complicated - you'd probably want to look into some kind of expansion board setup. But for the simple configuration you described, the Pico + LIS3DH + potentiometer should be all you need.
But even so, it's not a nice retail solution that you can just take out of the box and start using. If you can find an original KL25Z, that's a bit less work to set up. And there are a few actual retail options you can look at as well, such as Arnoz's boards.