I don't think Azor Ahai and PTWP are the same. Nor do I think AA is an unambiguously good character.

GRRM plays with stories a lot. Things that sound heroic or legendary in story are horrific when played out in real time.

For example, Lyanna abandoning her family for her beautiful prince sounds like a fairy tale. Sansa doing it is a nightmare.

More to the point, Stannis will have Shireen burned in the books too. He thinks he's AA. He thinks he will save the world. But it's just the act of a monster who has lost touch with reality.

Stabbing your lover through the heart isn't the heroic act of legend when it happens in real life.

The volcano will heat up the atmosphere melting the ice on Westeros.

Volcanos can cause warming over long periods of time because they release greenhouse gas emissions. But that takes many many years.

The immediate effect of a volcanic eruption is a cooling. The ash blocks rays from the sun.

I seriously doubt AA's big plan is releasing greenhouse gas emissions and then waiting a century or more for the earth to warm up.

https://scied.ucar.edu/shortcontent/how-volcanoes-influence-climate