well unfortunatly I do have some grout painting experience..first I'd want to know if the grout was a sanded grout or not... as it is a floor does it have a wide grout line or thin... like 1/8 " or so...wider lines are often sanded grout... and the size of the tile...to figure if it would be too much work..sounds like biggger tile though..
taking out the grout isn't all that hard ...a diamond blade in an angle grinder or roto-zip makes short work of it.. I recently used it to channel out some 1/2"grout lines line to run a water line to a refrigerator across a kitchen.. where they had a lovely big sub zero unit.. but no way of getting water it over to it... in an all concrete building.. with a passage way on each side... they had it like that for years as noone could help them.. one line of grout took 5 minutes to clear..digging through to the slab and then 1/4 ' into it..I ran the 3/8 " water line then regrouped.. for your regrouting just going 1/8 " through the old grout would be plenty to get a new grout surface to stick into...
as far as painting.. I laid a rubble granite floor here at home out of 3/4 to 1 1/4 " hunks of stone.. so it was deep and wide.. and needed a sanded grout..so I thought I'd use some.. but the color wasn't dark enough.. so I repainted with an thined out oil paint.. it wipped off the tile easily.. with thinner and hung into the grout fine for the last 20 years.. the grout will stain but you do need a binder in the pigment to get it to stay in there through washing and use..
often in cleaning up the origional grout they hhave left a slight sub surface radius.. you could try floating a tiny bit of a grout with an epoyx to just top this off...Lowes sells the epoxy kit for thheir grout.. but you can use it instead of water with any grout.. clean up quickly per instructions with vinigar/water... as it in the tinyest of amounts will stick to anything... almost forever...
have you got a close up picture of the grout line you have.. and one of the floor?
