How do you fix the problem when the old grout has turned gray and ugly?

I am renovating my bathroom. The old tiles on the walls are in good condition, but the grout lines are gray and I want to make them white.

I don't want to apply grout – it would fall off, and I was offered to paint with a special grout pen. Who has used it and can tell me if it meets my needs? I was also thinking of trying silicone caulk or silicone paint.