Yes and no. The brilliant aura is more of an indicator that these Pokémon are, on average, better than their standard counterparts in the wild. This is similar to the White Swirl in Pokémon Go indicating a Pokémon is boosted by the weather and has a higher maximum level on capture and higher base IV spread.
As Serebii as stated, brilliant Pokémon will have the following benefits:
- At least two guaranteed max IVs
- Guaranteed Egg Move
- Provides Watts
While they do lose their aura, they stay brilliant in the sense that these Pokémon have better chances of obtaining a better IV spread and do not need to be chain bred in order to gain specific moves.
Since you can transfer egg moves between Pokémon of the same species in this generation, you can potentially catch one with the egg move you want and transfer it over to your more optimized Pokémon (nature, IVs, EVs, etc).