... I mean, Garrosh rage gets triggered for many reasons:
When Garrosh gets hit by Varian, when Thrall said "you failed the Horde", when Magatha put poison on Cairne's blade, when Garrosh gets the thoughts of losing to Cairne during Mak'gora so much he breaks Cairne's spear, in all cases he was super angry, and yes... Taran'zhu talking about his father is another one. It ALL enrages Garrosh and makes his brain and blood boil.
You know what all of those events have in common? ... it's Garrosh's pride... it's not about Grom. The pride of Garrosh was in hurt in all those events.
In Taran'zhu's case, Garrosh thinks his father's pride is connected to his own pride. That's why TBC Garrosh was completely different person. Garrosh in TBC was ashamed what Grom did to the Orcs. What Grommash did was a stain on the pride of Garrosh. It wasn't about "trying to be loved by his father". He never sought for love, unlike people with daddy issues. It's all about the pride for Garrosh.
Once Thrall told good things about Grom, the pride of Garrosh was cleaned and he felt stronger than ever. Garrosh was not only proud of his father, but he felt no longer shame for being Hellscream.
It's not a daddy issues, I would say, "Pride issues". If you would say that Garrosh has pride issues, I would definitely agree without question. His pride is his weakness and caused most likely his downfall with everything. Grom (and all Hellscreams for that matter) plays a role in Garrosh fragile pride.