At the time the ability is considered activated (after all costs are paid), Rings of Brighthearth is no longer on the battlefield.
But that rule changed with Magic 2014
So if you do decide to use a Sail Fossil and an Amaura card in your deck, make sure to get a few energy cards on this card before you put it into the active Pokemon spot, or be prepared to attach three more energy cards on Amaura when in the active Pokemon spot to retreat it
Given how rarely the plan works out in my experience, it's as big a cost as possible
No risk of confusing players by showing too many background Pokemonits all alternate art, so the stakes are lower, and creativity is higher
In the past few months alone, millions of dollars worth of Pokemon cards have changed hands, with one particularly rare card smashing through the million-dollar mark in July of 2022