What sort of product is champion? I've never used it..
You may have to use both a pad and a polypropylene brush on your rotary, to cover all bases. The pad will work best on the flat parts but not get into the recesses, whereas the brush isn't the best thing for flat surface agitation but stands a better chance of reaching into those tricky little nooks and crannies.
Funnily enough I cleaned a tiled floor this morning similar to this. It was a glazed ceramic tile but formed to look like natural stone, with a slightly riven surface. True to form, the riven parts were gunked up.
I approached it the same way as with all jobs like this - use the most robust combination of chemical and mechanical that the floor can stand. In this case it was a floor stripper at 1:10 with a black pad. I elected not to use the brush after the pad, although as it dried I had to nip round with a hand pad and wet cloth to bring up some of the grout joints a little better.
One important thing to note about tiled floors and flagstones is that the dark stuff in the riven parts may have been there all along! It may well be the residue from where the tiler washed the floor over after grouting. Hopefully you've got a strong acidic product for grout restoration (mine is based on phosphoric acid). Apply some neat from a cotton bud to a very tiny part of the offending area. If it's groutl you should see it "fizz" and go white for a few seconds. If that's the case, you need to scrub the whole floor over with the weakest dilution of that product that you can use whilst still remaining effective.
The reason you have to stay on the lean side is that if you use the product too strong it can leave the grout looking artificially too light or washed-out, almost like an efflorescence effect. This can be counter-acted a little by rinsing the floor again with an alkaline product, but will still be evident to a certain degree. It does fade back after a little while too so it's not a complete disaster but best avoided.
