I have found it a revelation to learn the chord and improvisation way of learning a song.
Improvisation is the critical word.
I think what you mean is learning how to improvise rhythmic patterns that fit chord progressions.
Why don't piano teachers teach that?
Because most of them are preparing students for piano recitals where you're judged by some notion of a "perfect" performance. Or at least that's the mentality. Your pretending to be on the path of concert pianist.
Often that is mistakenly thought of as "classical" style. But, historically that is wrong. Musicians improvised in past musical eras. Decorating simple melodic lines with embellishments, improvising preludes, filling in the right hand of figured bass, those were some of the ways they improvised. Improvising a melody over a given bass was a big part of it. That's analogous to today's lead sheets of melody labeled with chord symbols. You improvise by embellishing a rough musical structure.
People talk about it as a "lost art" in the classical world but there is a lot of recent research and interest in the topic so it is being revived. "Partimento" is one of the topics to look up. It probably won't help much with learning pop songs, but you may still find it interesting. It might even throw some light on music you learned in all those lessons.
After re-reading the OP, I really think the confusion here is about accompaniment and musical textures in various musical genres.
look up the chords to my song of choice and improvise too and played the whole of John Legend's song ‘All of Me’ within minutes. Why do teachers have to focus on sight reading and musical theory?
What you improvised was your accompaniment to a melody.
You can do that with music that was written with that in mind. Homophonic texture is what we talking about: melody, with bass part or chord symbols, and chord based accompaniment as filler.
A jazz/pop lead sheet is such a thing.
A baroque trio sonata with figured bass is also such a thing.
Beethoven's Fur Elise is not such a thing.
Gershwin's Three Preludes is not such a thing.
Generally, solo piano music is not such a thing, because there isn't an accompanist. And to be clear, doing something like playing solo piano from a lead sheet is not playing a work written as piano solo music, it's essentially an on-the-fly piano arrangement of a song, a work for a vocalist and a band accompaniment.
Contrapuntal music (polyphonic texture) is pretty much categorically not the kind of music we are talking about. Bach fugues are not the type.
This isn't a classical versus pop music issue. It's a matter of accompaniment versus non-accompaniment genres of music.
If you took piano lessons and worked on figured bass realization or dance accompaniment, things like that, I'm sure you would have gotten lots of work improvising accompaniment.