Most of us knew an old lady that ccould play the classics of the fifties, Amazing Grace, How Great Thou Art, Tennessee Waltz, Do-Re-Mi. Music stores in particular were willing to teach piano customers how to "just play what I want to ". One meets a lot of heirs of these people selling off the piano or organ with a few folios in the bench, always the same ones. These were simplified arrangements, rather boring, and not usually very correct compared to the recorded performance.
The purpose of exercises is to stretch one's ability to include things that are not easy. There are many physical tricks to playing the piano. Correct hand position increases dynamic range, turn unders allow more difficult repretoire, repetitive patterns like arpeggios allow one to sight read more quickly. Furthermore a practice regime that includes some exercises stretches out the tendons and muscles for more advanced stuff, in the way that runners do stretches before they run a race.
Nobody I know used Hanon, that seems to be a European thing, but some famous players around here used Edna Mae Berman exercise books and later Czerny, as I did. The purpose of the teacher is to point out the physical techniques one is not doing correctly, and the expression and physical points one is blowing by in the passion to get the melody down. A good teacher also assigns repretoire that the student didn't know existed and can't yet appreciate.
Another thing a good teacher can do is modify the standard technique for the student's particular body type and hand conformation. Perhaps in racially uniform Eastern China they can give classes to 300 students at a time on piano (I saw a hideous picture of 300 students at 300 pianos in one room), but in the USA we are such a mixture that students have radically different dimensions of their body parts. My body type is particularly unusual, and my teacher was able to find me pieces that I could excel at easily compared to the average student. She also did not rant on about standard practices, like never putting the thumb on the black notes, the way the printed fingerings in the student arrangements are always done. Watching some Russians play in close up video, no wonder there is a phobia about thumbs on black notes- their hand dimensions are totally different.
So if you want to be more amazing in a few years than the standard old Grannie, find somebody who will stretch your abilities to things you never knew were interesting. Don't neglect the standard pieces, pop music in particular is a better aural teacher of chord theory versus melody than my teacher ever appreciated. But my teacher taught me many useful things I never would have thought of.
Topic: Practing songs vs. doing excersises