If the dragons truly were poisoned by the maesters, is it possibly that Aegon III may have supported them in this? He was said to have been terrified of the dragons, and after the civil war that left him an orphan, I can see him viewing the dragons as a cause for destruction.

I doubt it. Aegon III might have personally been afraid of dragons, but he understood that dracocracy had been the foundation of his House from the beginning. 

All the evidence we have points to Aegon going to some lengths to bring them back:

“Did we learn nothing…from the nine mages?”

“Nine mages crossed the sea to hatch Aegon the Third’s cache of eggs. Baelor the Blessed prayed over his for half a year. Aegon the Fourth built dragons of wood and iron. Aerion Brightflame drank wildfire to transform himself. The mages failed, King Baelor’s prayers went unanswered, the wooden dragons burned, and Prince Aerion died screaming.“

“Yet together, Aegon and Viserys ably dealt with the remaining turmoil in the realm…they even attempted to restore the Targaryen dragons, despite Aegon’s fears—for which none could blame him after witnessing his mother being eaten alive. He dreaded the sight of dragons—and had even less desire to ride upon one—but he was convinced that they would cow those who sought to oppose him. At Viserys’s suggestion, he sent away for nine mages from Essos, attempting to use their arts to kindle a clutch of eggs. This proved both a debacle and a failure.”

For me, it works much better for Aegon to be genuine in those efforts – it’s the politically smart thing to do, as without dragons the Targaryens were doomed to diminishment; narratively, it’s a great case of the human heart at war with itself. Aegon steeled himself to overcome his worst nightmare for the good of his family, and still failed. No wonder they called him the Broken.