Even with today's modern medicine and technology, anyone suffering the extensive burns that Aegon did would have a small chance of survival. Yet with basically medieval level of medicine and poultices, he survives????? Stretching the suspension of disbelief just a bit far methinks.
The stretching that occurred with that event is still minor compared to the stretching out of what should have been a couple of episodes into an entire season. Even the filler in House of the Dragon has filler.
So in a show with Dragons, White Walkers, Giants etc., it is their burn medicine that you don't believe in... This is a fantasy show it doesn't take place in medieval Europe:-)
Even fantasy settings benefit from verisimilitude, or a sense of realness. I think the issue for OP is that there isn't much to imply that Westeros has any advanced scientific knowledge, or that they employ magic to heal injuries. So while, yes, dragons may exist within this setting, people still receive injuries and have limited means to recover from them as far as the audience is aware. In contrast, Beric Dondarrion's resurrections from R'hllor weren't questioned because it was obviously magic/divine intervention, so there wasn't any internal logical inconsistency.
Traditional wound healing medicine is superior to modern medicine both in terms of cost and in terms of efficacy. Every medicine we have today is inspired by plant molecules modified so that they can patent it and sell it more expensive. For burns there is no other choice but to apply poultice rich in Vitamin A and E, like rose hip oil or other ointments rich in those vitamins. There is nothing much you can do. What would be surprising would have been if he received reconstructive facial surgery.