You don’t know anything until you try to take them apart. I would pull out all the screws on one and see if the stair comes up. If so, glue with construction adhesive. If not move on to countersinking the screws and plugging with oak plugs. They will be far less noticeable than filler. Make sure to match the kind of oak and you want a plug like this that is face grain. Looking at the current screw it looks suspiciously like a drywall screw so it wouldn’t surprise me if some are snapped. You definitely want a strong screw like a deck screw.

The top step has nine screws on top and the bottom tread has none. Looking closer, most look like they do have nails along the back and 2-4 screws along the top front of the tread. I don't think treads will come up.

I was able to access the area underneath the stairs to see more. This was likely installed in 1963... pre-Liquid Nails. They don't look like drywall screws IRL.

Would removing the screws going through the risers into the back of the tread help alleviate the squeaks, or is this standard? Is it strong enough structurally if the tops are nailed in (from top and/or back) with no screws or adhesive? We have four seasons here so lots of expansion and contraction in our house. It seems like replacing any screws that need to stay with a good deck screw is a good idea.

I suspect those screws were not there originally. They were put in at a later time, possibly before the stairs were carpeting, to try to take care of stair squeaks. You may open a big can of worms, if you remove the screws, since they were obviously put there for a reason, unless you plan on replacing all those stair treads. I would sand, and refinish, and then put the runner down the middle of the stairway.

Wood steps are slippery, ask me how I know. They can be real arse busters. You will need carpet or rubber treads at the minimum to prevent an accident.

@HALLETT & Co., I'd love your opinion.