When does it make sense to use a grout pen?
Use a grout saw (manual) to remove all the loose/failing grout and re-grout with black (or dark-colored - blue perhaps) grout. 



White tile with white grout is dull even when it doesn't have issues. Dark grout also does not require the cleaning effort that white grout does. Tile is easy to clean, grout less-so. Here's a handy image of white tile with different grout colors.



I don't believe the "miracle" product will do anything useful for grout that is eroding. This should be a "one day out of the weekend" project, it's not a remodel. If you have an alternate shower/tub to use it could also be an "hour or two a day" project for a week, unless it's a gigantic shower with tiny tiles.

Looking into this a bit more I found several mentions of having no effect on mould, other than when you next scrub the mould, you have flakes of white paint coming off (since that's all it is.) Indeed, anti-mould spray was recommended for cleaning the grout. But given that you already have the grout failing I stand by the advice to replace it.