My teacher told me when I'm learning a piece I have to understand the underlying harmony, only then I would be able to learn and play it easily, he also told me to never learn the notes itself (because every note is part of a chord or harmonizing with a chord I could only imagine) , instead figure out the harmony then play it over and over again until you internalize it, then play the actual notes that the composer written and see how they take a chord progression make it into a song.
Though that's valuable advice, I'd be careful with taking it too far. A chord progression is one possible abstraction of the actual notes played, and having an idea what the chord progression is can give you a useful way to understand the song. But it's not the case that composers always take a chord progression first and then make it into a song; sometimes composers will write without thinking about chord progressions, and some songspieces don't have clear, identifiable chord progressions.
Think of the ability to identify a chord progression as a very useful tool, but not the tool for every job.
so in order to be able to identify these harmonies fast, I have to basically internalize all the scales, (most of the) chords, their chord degree, and obviously knowing what the chord progression sound like
It's daunting and I don't even know how to start internalizing at first, I tried to memorize the chords right?
Cmaj C E G Dmaj D F A
but then I have to be able to immediately know their inversions and Nashville Number System
so Cmaj 1,3,5 C,E,G plus inversions. it's not easy for me to be able to switch them around and to be able to know them as their inversions so it looks like I have to internalize each and every inversion.
I don't think you have to worry about learning that in a big 'block' all at once. I think the most useful way to learn that knowledge in context is to analyse a bunch of different types of songs in different styles. Writing down the chords with a number system is very useful - it allows you to start seeing common patterns and associating them with the sound you're hearing.
What notes are in chord Dm?
What's it's V/V or iv/V chord
What's the relative Major key of A major
What's Am first inversion?
If I want to modulate from C major to the V/V chord what do I do?
having the answers to these questions immediately on the tip of your tongue isn't necessarily as important as understanding what they mean and being able to work out the answers using your knowledge of the underlying patterns of music. If you get into the habit of analyzing and writing music regularly, you'll fairly quickly come to be able to work out all the above.
I believe that internalizing these foundational pieces could turn people in true musicians, people should be taking as little time as reading and understanding tbh, any time wasted on figuring out the harmony is wasted....
Maybe... but from your post I somewhat get the impression that you want to take a shortcut to 'musical' knowledge by learning a bunch of abstract patterns and thinking you'll then be able to apply them practically to music. Of course you can identify some generally-applicable patterns, but I'd recommend internalizing the knowledge by doing as much analysis of real music as possible - that way you will be learning the patterns in context, seeing how they really apply, and also perhaps finding some places where they don't.