Vinyl and Linoleum Flooring - Why no resilient on stairs?

I have a raised ranch with a side door entry. Once inside, you're standing on a 4x4 landing and must take 10 steps down to the basement or 3 steps up to the kitchen. I'm about to redo the kitchen and have settled on sheet vinyl for the floor. The challenge is those steps, which are currently covered in the same 12x12 vinyl stick-ons as the floor.

Everything I've read said resilient vinyl is not recommended for stairs but there's never a 'why.' Only one site vaguely said, 'for safety reasons.' As this isn't a full flight of steps, just three, and there's always a rug on the landing for feet-wiping, is it really inadvisable to tape or glue down vinyl on these stairs, with a good nose edging?

While you will use the stairs 99% of the time, a relative or other guest, especially an aged, infirm, or inebriated one may slip and injure themselves. Guess who gets sued.