Use the butt end of the handle of a large screwdriver. Place it on the mound, quick tap with a small hammer on the tip. Et voila!

As always try it first in a non-conspicuous area..

As with my luck you push down in one spot and half the playfield flies up and hits the ceiling fan which in turn shoots across the gameroom and takes out the backglass of a Fireball.

Yes, and a nut driver is even easier to hit squarely.

You wouldn't have thought of simply banging it down flush? Even a rounded hammer face will do it directly. You can put a piece of cardboard or cloth over the raised hole first to protect the graphics. Do you own any tools?

Also, to repopulate, it's best to fill the hole with some wooden toothpicks dipped in white glue (just break the top end off above the pf) or, optionally, drill the hole through and use a long screw with a t-nut or regular nut and washer underneath the pf.

No, toothpicks are used. Say you are reinstalling a screw but it still turns freely all the way in.
You place a toothpick in the screw hole so the screw has more wood to bite into.
I use this technique all the time.

+1, same here. But to the OP, yeah no need of filling the holes in unless they are stripped because you will be using them again.

For stripped out holes I cut pieces of zip ties to the depth of the hole put a small dab of wood glue and stick them in the hole. the nylon grabs on to the screw and gives a snug fit and you can take the screw out as many times as you want and it wont strip again. Obviously it will if you torque it down way to tight but we are talking realistic tightness here.