Mop disintegrates when used on coarse stone floor - any good alternatives?

I had a black slate floor just like that. I cleaned it 3 times by hand as soon as it was down and dry - still not used then treated it with 3 coats of oil and let it soak in. Ever since it is easily cleaned with a mop and liquids don’t sink in.

Use a stiff bristle brush, or a wet broom.

Depending on how filthy, scrub with a brush and wet vacuum until you've rinsed it clean. If not so bad, sweep the dirty water to a spot where you can sponge it up without scrubbing the sponge (or use rags similarly - pad, don't wipe.)

A sponge mop would be hopeless, as you've found.

A (cotton) string mop should work - it might shed a little lint when new but is much tougher than sponge or fabric strips.

There are also cotton-covered mops similar to the sponge type, but I can't seem to find a search term. The fabric can be rather tough, which I think is what you're looking for - imagine (or test with) the leg of an old pair of jeans over a sponge mop.

Depending on how dirty it gets, and how willing you are to clean up the odd dirty spot with a cloth, a steam mop might work. The bonnet on the one I had (similar to this) used much shorter fibres that didn't snag, and it made the floor come up nice and clean. The downside is it's expensive to try out if you find it doesn't work too well.

As an aside, if a few odd sharp spots are particularly bad (snaggy or obvious) you may be able to take the sharp point off with set/dry sandpaper (silicon carbide paper) by hand, assuming the surface isn't sealed.