Sometimes I wonder if we teach music wrong.
I was fortunate to have an elementary school music teacher who used the Kodaly method. I learned folk songs with movable do solfege, and banged on xylophones with only five notes, before ever getting into sheet music and instrument-specific skills.
I have been really frustrated.
Yes, it is frustrating to struggle with skills that other musicians take for granted. I struggled with reading bass clef and develop hand independence. I had never needed these skills as a woodwind player, and have been reluctant to put in the time working with beginner pieces to develop them, when I can do other things on accordion at an intermediate level.
I think we have established that singing, humming, or whistling is an essential part of how people develop the listening and memory skills needed to play by ear, but you are right that most singing teachers are working on an entirely different set of skills. Last year I bought some singing video lessons to work on things like lip bubbles and vowel modifications and breath support, and it's helped me improve my tone and crack fewer notes when navigating the passagio, but none of that is relevant for learning a melody by ear!
I can sing along to some tunes. But what you are forgetting is: they might be perming it in a bad key for your voice.
Not important! Once I've "learned" a tune by attempting to sing along, I can transpose it to a better key for my voice. In order to do this, you will need to encode the information as relative pitches (something like
movable do solfege) rather than "muscle memory," i.e. "here's how my larynx feels on beat one of bar 2."
Copy on my instrument, or by singing? No. I've got one chance in twelve of pressing the right button on the first try.
If you played folk songs on melodeon, you could improve your odds to one in eight! On a fully chromatic instrument, warming up with scales can serve the same purpose. But even so, there's a lot of trial and error involved in learning to play by ear.
My guesses are more accurate than they used to be. On piano accordion, the sequence of skills acquired was something like this.
1. I can tell the next note is higher, so I'm going to press a key to the right (or down) from my current note.
2. I can tell it's a small interval, of the sort I hear when I practice scales, so I won't stretch very far
3. I can tell it's a small interval, and doesn't have that dissonant or dramatic quality I associate with accidentals, so it's probably a B rather than a B-flat
4. I can tell it's a bigger interval, so I'll need to skip some of the notes in the scale. I'm not sure how many though
5. I can tell it's an octave because it sounds like other tunes that use octaves. "Chesnuts roasting on an open fire" etc. I know how much I need to stretch for an octave.
6. I can tell its a medium size interval, so I'll stretch about so much. 50/50 chances whether I get it right
7. I am 75% sure it's a perfect fifth, and will need to stretch exactly this much
8. I am 95% sure it's a perfect fifth, and will need to stretch exactly this much.