Looks awesome. Joust is on my list but its going to have to be a restore candidate like you found. Your making excellent progress. Be sure to stop and enjoy one and a while!
As for the screws that broke and are now stuck. I have an extraction kit and you may find this to be a fairly good solution depending on how far down they broke. What I have is designed mostly for removing stripped screws where you have a bit of the head left. Clearly this is not the case for you but the principal should none the less apply.
My kit consists of the following two pieces you use in series. Step 1 is the take a drill bit with a very shallow taper and cut a cone into the top of the stripped metal. Step 2 is you basically take what amounts to a reverse threaded die and start screwing it into the depression using a power drill in reverse- when the die hits bottom the threads grab and bite the metal and you back the whole thing out.
The issue is going to be two fold- can you actually get anything to hold center on the broken shaft and cut a shallow depression. If you can figure that out your golden and just need to find a small die that matches.
Anyhow- not saying I have done this for a broken screw and it could be extremely difficult given diameter of screw and the fact that you may not be able to hold center. However- you could use an oversize bit (like 1/2 inch) and take the top 1/8 inch of wood off from the screw hole on the underside- that would hold center over the broken shaft and generate a nice cone to grab with the die- if the screw shaft is up that high if the break is down the hole you may be up a creek as you could not do this without damaging the Playfield.
Then its just a matter of filling the small neat 1/2 inch hole and sanding and repainting the underside. I suspect your going to be filling a few damaged spots and painting this anyway so it may be worth a try.
Granted drilling a 1/2 inch bit into the bottom sounds sketchy but with the right bit I really think this could work and its not going to effect anything at all- filling is trivial and you can fill with a compound that will hold the new screw even better than wood if you wanted to so no loss in integrity if this works.
Advice- break a few screws off in a 2x4 and test this method before you try it.
Else maybe someone else stops by with a wiz bang trick and we both learn something new.
Good luck and I hope this makes sense.