Removal method is chosen for the coating and the stone, and tested before being applied broadly. Chemical stripping suits some coatings; mechanical removal suits others, and on softer stone the wrong choice damages the surface.
The stone is brought back to a raw, uncoated state so it can be assessed honestly. Only then is a restoration method chosen, because until the coating is off nobody can see what they are dealing with.
Residue in joints and at edges is dealt with specifically. Stripper that has run into grout lines and dried there will show through any subsequent finish.
What replaces the coating is a deliberate decision. Often the answer is a mechanically polished stone surface with an impregnating sealer, which cannot wear through in the way a topical film does.