Just buy pye grout, mix with water then fill in the gaps.

Then wait for about 40 min, use wet cloth to wipe away excess grout.

Easy. If want all the grout to be of same color there are those paint pen type with fungicidal formula.

Not really. From my experience the pye grout will stick to it quite well.

Oh yea. Better still, can pour in pye krack seal to all the grout places and let the liquid seep in.

Cracking grouts are a sign of old building and likely the waterproof layer also likely to have cracks. The krack seal will flow in and form a waterproof layer. Do this before applying grout as pye grout is waterproof.