If the existing treads are not dado'd into the skirt boards you can remove and replace them. But you need the rise to vary no more than 3/8 of an inch over the staircase to meet code. Rise should also be no more than 7.75 inches. These are the requirements in my area, yours may be different. Treads are typically 1 inch and construction lumber 1.5 inches. So you could be changing the rise of the bottom stair by .5 inches. If the others are consistent, the only other one that could change is the top.
If the existing treads are dado'd or the rise if going to vary too much with replacement treads, you can cover them. I did this two houses ago. I used 3/8 inch thick flooring, cut off the nosing of the construction lumber treads, glued and nailed on new oak nosings, and filled and painted the low grade plywood risers. If you go white risers with clear finish treads, you can do a lot of caulking and filing of small cracks without it being noticable. Clear finish risers and skirts will result in a need for more accuracy. I cut the nosing off with a circular saw and handsaw. I did not have have an oscillating saw then, I would do the ends with it now. It was messy and not terribly easy but we liked the end result.
We had and oak stair case in that house, the next one, and have two in the current house. We had no problem with slips and falls. I'm not saying bona high traction isn't a decent idea, just several decades in my houses haven't been problematic. Handrails have saved us from any falls when somebody looses their balance. I put normal poly on all these staircases.