Agree that is the wrong product for that application. The installer can cut and make a return end(this example is opposite your problem, but is done the same manner). also, were I doing that job, I would install some kind of support behind that nosing, since vinyl is easily cracked/damaged. And some backing(non expanding foam or foam sheeting cut to fit) will aid in installing the return end piece.

I forgot to mention the third option, which is to use real wood stair treads and stain to match the vinyl. However our house has several "step downs" and we have to use stairnose trim on those.

I'm assuming you mean vinyl on the risers (vertical surface of the step)? Vinyl is a very different beast from laminate. Vinyl is flexible and wobbly. In theory, the risers should be FULLY adhered to the substrate of the riser. If they are not, then the adhesive is not 'full spread' on the back.

Nature of the beast when using vinyl planks and stair nose. There have been several threads in the past on this subject. Anyone contemplating wanting to put vinyl on stairs needs to carefully look at the system offered by the different companies when you're choosing the flooring. I plan on putting down vinyl plank. My stairs have solid oak treads (under carpet). I plan to finish the wood and live with the mismatch. No way am I going to start hacking the nosing off of solid oak treads and replace with vinyl. Probably will end up using a carpet runner if the difference is too stark. Because I know solid oak is going to outlive vinyl by several factors.