How to repair a closed-stringer staircase

Sounds like you want to use contact cement. Make sure both surfaces are clean, apply contact cement, let it set up, and then put the rubber onto the wood.

It depends on how the particular stair was constructed. Although the absence of a central stringer is evidence that the treads are capable of spanning without the support of a riser, it is not proof that they are adequate for the span. Likewise, the absence of a central stringer does not mean that the stair is laterally stable without risers.

Even assuming the stair is constructed to withstand the removal of the risers, the feasibility of 'simply cutting the tops of the stringer' has a lot to do with what constitutes feasibility. Risers stiffen. Bouncing treads or squeaky steps are not consistent with the intent. What performed well before, may not perform well after modification.

In addition, aesthetics probably plays a role and might even be the driving motivation. Just having open risers is unlikely to be enough, they need to look good, and the ease with which the desired results can be obtained is probably also a consideration. Sawing and cutting and patching a closed riser stair to the point of an aesthetically satisfactory open riser stair is likely to be extremely labor intensive. Particularly given that errors are part and parcel of the learn as you go method one must use when tackling a problem without a straight-forward solution.

So sure, it might be feasible. That doesn't necessarily make it cheaper or easier or more cost effective than simply replacing the existing stair with something close to what you actually envision. Cutting out the risers doesn't change the basic fact that it's a construction project where quality of fit and finish matter.