Manufacturors have to pass stoves to burn on a decorative hearth only.. best way to search is the whatstove website, advanced stove search, and check the 12 mm hearth ok box, that will list all the stove which comply..
And yes, in practice it can look very nice. WARNING many manufacturors photos show stoves installed contravening a multitude of regs, am bored of telling customers they can't have it installed like in the brochure, it won't comply!

I suppose you could seal the edges of the glass to the floor with clear silicone sealant to stop the ingress of dirt.

I have just installed a 6kw wood burning stove and one of the manufactures recommendations for a noncombustible hearth was glass - I put in tiles as the hearth, with the remainder of the floor as engineered oak. I guess glass would be OK as it seems that a number of manufactures suggest/recommend it but make sure that the floor is flat and that there are no sharp points or protrusions on the stove feet to avoid point stressing the glass and causing it to shatter.

What Finny said.

This part of the Approved Docs confusing in my view but the end result is quite simple. Basically If the stove is certified as incapable of raising the hearth temperature above 100C then you can get away with a thin glass/granit hearth rather than a very thick concrete hearth. Even so the thin hearth must meet certain minimum dimensions (eg it must extend out in front and to the sides of the stove by a certain distance) and must be at least 12mm thick. The reason for that is to form an step to discourage people putting a rug too close.

I wanted a slate hearth but was warned that it scratched easily so opted for a matt black granite. Went to a local stone mason and he had some 20mm glossy black granite left over from a kitchen work top project. I made a template of the recess and added a bit out front to meet the regs. He cut it to size and honed off the gloss surface to leave it matt. Gave it a coat of sealer before putting the stove in.

One thing to watch is that it's tempting to stand on the hearth when cleaning behind the stove so make sure it's s well supported and thick enough for that. If there is a recess the glass may end up T shape like mine and the corners look vunerable to being stepped on. Not that we've had a problem.

I like the idea but a hard impact from a solid object could mean end of hearth. Other materials would chip. You could end up with a kit form hearth and lots of little decorative glass pieces.......