Q:

How do I manage internalizing foundation for chords, scales, hamonies and voicing

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
I really like this idea because it allows you to learn the piece quicker and you won't get bored of the song by the time you finish learning the whole piece.
My goal is to be able to recognize harmonies quickly, for example, these and knowing them by Nashville Number System so it'll make transcribing and communication easier)
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?
Do you guys know what I'm talking about? (I don't think there are enough examples for all the general situations but I hope you understand what I'm after).
Some pieces' harmony are crazy, sometimes when composers are really feeling it, they tend to harmonize the chord progressions with chords in the relative key, the minor key.
Some composers even use the degree of the scale as chords like The IV of V, The V of V. they add seventh chords in order to modulate to a different key, you'd dread this process if you haven't internalized the scale and chords because you'd be taking hours and hours just figuring out the chords, even if you got the chords, without the internalization, you wouldn't be able to understand it fully. 
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 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....
Okay maybe not that fast but if your taking time to go through inversions or scale notes or using a reference, it's mostly waste of time
I'm seeking for advice and efficiency and I need your help to be able to achieve this as I know some people might have gone through this. I know that taking time to learn this entire system could lead to too many trials and error
Thanks for reading my post 
EDIT: I just wanted to clarify
 I'm going to give an example
check out this video 

0:50 he could immediately identify that the
F-7 is the vi chord (Because F is the 6th scale degree)
Bb-7 is the ii chord
Eb7 is the V
Abmaj7 is the I chord
in the key of Ab
I'm assuming if he had lead sheets in all 12 major scale he could know which each chord are which in relation with the keys
He says that he made a chart for this but like I said if your taking time to look back and fourth on the chart, it's wasting time and that's just identifying it what do you think is better? besides does he need to even think or refer to the chord chart? This is what I mean by Internalizing
continuing on the video he explains that the diatonic chords are chords that fits in the key but sometimes the chords break out and become chromatic harmony so basically I have to be comfortable in any key and be able to know what the composer did
like did he change keys from Ab to relative minor or does he treat V chord as one?
At 3:21 he knows immediately that G7 is not part of Ab
so he treats the G7 as V cause he want to modulate into C
4:42 he is able to tell that it's the exact chord progressions because he is familiarized with all these keys
watch till 6:02 so you see how effective this is by understanding the harmonies, want to internalize the scale, harmonies etc

A:

It seems to me that you have a teacher who has good intentions but it bad at actually teaching. What you are being asked to do is all very useful, but you need to be taught how to do it! Nobody (except the likes of Mozart) just figures all this stuff out for themselves.
The type of questions you are asking are all covered in a typical "theory of music" course - but expect to spend a year or two learning the material to the level where you have internalized it, not a few days. Your teacher should be taking you through that sort of course in a structured way, so you don't get overwhelmed or bored, and relating what you are learning about theory to your practical work as you go along.
You don't say what instrument you are learning, but if you are a keyboard player, when you get to the level where a singer gives you some sheet music you have never seen before, which happens to be in G# minor, and says "that key is too high for me - can we do it in Eb minor instead," (and the singer means "right now, not after you have taken it home for a month to practice") you most definitely need to have internalized everything in your OP!!! 

