If gaaiginōn came into Old French via Old Provençal, the hard /g/ could have survived.
But how could it have entered Old Provençal without it being Frankish in origin? Old French is kind of... in the way between them. Of course it's possible that the word came from Alemannic instead. But is it not possible for the g- to have survived? After all if garir has g-, why can't this word?
Ok, I see. But still... if w > g survived, then g itself could have survived too. Are there any examples Germanic words with g being borrowed into French with j?
When I say /w/ > /g/, I actually mean /w/ > /gu/ > /gu/, /g/. A really good example is gay, which came into Old French from Frankish as jai, but was displaced by gai via Old Provençal, which was from Gothic gaheis, both ultimately from the same PGm source.