Your mistake is in thinking of melodic and harmonic minor scales as being keys. You are also confusing aeolian modal pieces with minor key pieces. A song in a minor key can use any of the notes in melodic minor, harmonic minor, or the aeolian mode of a major scale. If you don't take at least one note from harmonic minor, you can't get a dominant V chord in a minor key. Without a dominant V chord, you are dealing with a modal piece of music.
So yes, actually, those chords are all in the key of A minor.
Harmonic and melodic minor scales are just ways of explaining certain Bach pieces, and they have little relation to real music. They were made up after the fact to explain music which was written before the "rules" were ever codified. The more you learn about harmony and music theory in general, the more you will understand that there are no rules, just understanding. This is why I always find the people who claim they don't want to learn about theory because it would "limit my songwriting," to be such idiots. They almost invariably make music which conforms to the most basic ideas which you would learn in the first month of harmony training.
Music theory is not about rules, only understanding.
Light
"Cowards can never be moral."
M.K. Gandhi