I want to understand how ear training to identify different intervals helps me figure out what notes are in a song, and what key it's in.
Working with a purely. or nearly purely diatonic song, yes, it will be of use. But not just on its own.
Knowing major and minor thirds will identify which sort of key a song is in. Providing one can find places where the song feels at rest, places where it could finish, leaving one with that 'ah, we're home again' feeling.
Knowing what a leading note is likely to do, and aware that it's a semitone from the root, will help identify a key when there's a perfect cadence. As will knowing what a P4 sounds like, in the same circumstance.
There aren't that many intervals to get your head round, and generally, if you understand the main ones, your playing will improve, initially thinking along the lines of 'the next note is a P5 from where I am now, so it'll be...there, on this instrument.' And this mindset becomes automatic eventually.
I use it in lessons. Play a short (initially) phrase, in a specified key, and the student will map it out in their mind, and play it back, without much trouble.