I don't see a reason to spell it phonetically when it is a real word (forge in esperanto). A phonetic spelling would also only be more digestible to readers who know the language the phonetic spelling is tailored at (phonetic spelling is language specific as different languages use different ways to represent different sounds).

ĝ is simply the english sound of the consonants in the following words: "john", "gem", "jar". And j is pronounced as the y in "yes" and "yoink"

The diacritic would clear up confusion, because "g" without the diacritic has different sound (like the g in "gamma", "girl", "go" in english). The diacritic as a bonus would also makes it clear that it isn't supposed to be pronounced it as if it were in english, because english does not use the ^ diacritic. It would also extinguish my annoyance at seeing a misspelled word being used as a trademark.

https://en.wikipedia.org/wiki/Help:IPA/English

This is phonetic^1^ spelling. The only good one.

^1^ Actually phonemic. Don’t kill me

There's no universal "phonetic spelling."

Every language and its user have unique accent and they will intreprete phonitic spelling differently.