Drywall – How to finish drywall under stair treads and risers seamlessly

I don't think sanding the surface will produce a surface good enough to paint. It's certainly a lot of effort for uncertain results.

If the underlying surface is very rough it might still show through the wall paper, unless it's very thick (1000 gsm lining paper for example) or you do some sanding.

The best answer might be to continue to peel off the paper and then skim the wall with plaster. Depending on your plastering skills you might want to get a professional in.

The result should be good enough to paint.

Though you might be able to mud over them, you're looking at 1-2' of mud (feathered) around each screw to make it not noticeable, which basically means you'd be covering the entire piece of drywall with mud. Like pretty much everything in construction, it will be much easier to do it properly now rather than try to fix it later.

If your driver is slipping out, you are probably not using the right bits (or your bit is worn out) or are not pushing hard enough on the drill to prevent the bit from slipping out. Make sure the bit fits well in the screw head: it should not have any room to wiggle in any direction, and should not slide around. #2 Philips is the usual here, but depending on the screw head it may need to be a sharper/longer or blunt tip.

It could also be that your screws are not the right type. If you're going into wood studs, generally you should use coarse thread screws. If you're going into metal studs, it should be fine-threaded screws.

Since the board is also 5/8, it may be that the screws are not long enough. Try longer screws, which should get a better grip on the studs behind so once they start going in, they'll go in as deep as you need.