I think you’re right
@annieeg, when you took the screws out, the selector mechanism has dropped and the screws which hold it up can’t catch on to it, to relocate it.
If it were me I would be tempted to remove the base plate or rather ask DH to remove the base plate as I am married to a man who can mend just about anything.
With the base plate off it might be possible to push the selector mechanism into place so the screws can reach it to hold it in place.
Please don’t do this on my say so, only if you are confident with this sort of thing.
If you can get it back, give the air threader tubes a good flossing with wire, always pushing or pulling any debris towards the needles.
As long as you have the screws etc a service engineer shouldn’t have too much trouble getting things working again.
Let’s us know how you get on.
Can we please have a fingers crossed emoji?