Any modern engine does this automatically on PNG import, or as part of material/shader setup. You want different formats for different things, e.g AO, normals, bcs different formats have different compression artifacts.
How much extra time does it add for startup? For a single texture it’s probably negotiable but for a modern game, we’re talking about thousands of textures, so maybe it adds up?
I assume parent means "import" as in "asset pipeline import", not importing at game start. Using hardware-compressed texture formats will actually speed up loading time since you can dump them directly into a 3D API texture without decoding.