Playing by Ear - from the Top Down

If you can recognize 2 things when you're listening to music, you will be able to play any song in any key, without looking at music.  Surely that's what people mean when they talk about "playing by ear".   Here are the 2 things:
   1.   Chord Type
In their basic form, there are really not that many kinds of chords to recognize:  Maj, Min, Dominant 7, Diminished, Half Diminished.  It can be a little challenging when the root is not on the bottom or there are a bunch of fancy jazz extensions and alterations thrown in, but it is a skill that can be learned and you will have to learn to see through the clutter to the basic underlying chord.
   2.  Degree of the Chord on Top
Every melody note has a subjective quality that has everything to do with how it relates to the underlying chord.  For example, a note that is a b9 of a chord has a distinctive tension and sound that it would not have if it were the 5th of a chord.

  Once you have those 2 things in mind, the chord becomes clear because there's only ONE chord that satisfies those two conditions. It is possible to get very fast and making the decision about what to play and it can be done as you're playing.  This is not another academic exercise but a practical and fundamental hearing skill.

In the 8 bar section, below, of We'll Be Together Again,  I have spelled out these 2 characteristics along the way, characteristics of the quality of sound that can be heard and recognized immediately.  BLUE is the chord type, RED is the degree of the chord that the melody occupies. In each case, their is only one chord that will satisfy the two requirements and playing the tune by ear is simply a matter of putting it together.    To me, this way of hearing through a tune is far easier than trying to hear ( or worse yet, memorize ) root progressions.  Note - on the second chord below I've also often heard that as the 9th of a dominant7 chord  (which means it would have to be Gb9) - I can't say I know for sure which is the original chord without some research

A Simple Way to Hear Chord Changes to a Song

George Shearing - Blindness Forces you to Play "By Ear"

The Performance Version, with Chords derived by Hearing Top Note Qualities and Chord Types