The dark line is just the result of exposed tile base material color, not due to heat or anything. Painting over it is not good idea because for table top it is exposed to hot and cold pot and bruises on daily table top routine. It should be just slightly grind just to make it not too sharp edges.

I'm afraid that there is nothing much you can do about it unless you can find something like edge protector but it will add some thickness and cause dirt accumulate along its line. I'm not sure if there is any contractor that can do precise polishing with some guide tool that will ensure constant angle and depth that will leave the base color looks straight.

Looking at the picture, if the polishing guy was using his steady hand, I'd say he has done very fantastic job already