Tile Grout Removal Tool Dirty Grout Removal?

Dirty Grout Removal? - tile grout removal tool

I have ceramic tiles in the bathroom with dirty grout (probably not closed properly) actually change the color to a dirty brown color. I experimented with a Dremel tool and bit to remove the mortar. I can adjust the bit depth of the grout and remove the surface dirt is enough to show clean the mortar. Can I stop here and that's what I do or not, dig into the mortar and grout seal and RE? What is the approximate depth of the grout? Thank you!