First off, let's define things a little. When composing a piece using exclusively natural minor, we would refer to this as being Modal in nature; when using melodic and harmonic minor at times and natural minor at other times, it would most often be considered Functional in nature. When you find exclusively harmonic or melodic minor, we could make an argument for it being either Modal or Functional depending on how the piece utilizes the tonality.
The choice of using exclusively natural minor or including harmonic minor and/or melodic minor is basically the choice of the composer. There are tendencies based on genre, so you may be noticing some patterns as a result of that. Most Classical music will utilize harmonic and/or melodic minor (earlier Baroque music tends to be more modal and some Romantic and Modern era music also used a modal approach) and is basically unrestricted as far as what key the piece would be in, so if you looked exclusively at Classical music, I would guess that your thought process would change a bit. Most Rock music, by my estimation, tends to be modal in nature, as well as being guitar based, so you're more likely to see standard guitar keys (A, E, D, G, C, etc.) using a modal approach.
The main difference between modal and functional approaches has to do with how the chords move and work within the piece. The initial intention of harmonic and melodic minor was to add a leading tone to the V chord to increase the feeling of tension and resolution. This approach typically uses natural minor throughout a lot of the piece and switches to harmonic or melodic minor when the V chord is being played. Modal minor tends not to use a V-i cadence in this way. We tend to find different cadential figures in these pieces, such as bVII-i. These two approaches can have very different affects on the overall feel of the piece and depending on what you're going for, it could be considered inappropriate for the genre to do so, for instance, if you were trying to write a piece and sound like Mozart, you would want to use a Functional approach, or things just won't sound like Mozart.
When a piece is using exclusively harmonic or melodic minor, it's possible that it is functional, using a V-i cadence, but more likely it would be modal. This sound is often said to have an eastern feel and often employees a drone or pedal tone. The use of a pedal tone on guitar encourages the use of open strings, so you would most likely run into A minor and E minor, as they are the lower strings and provide a thicker tone for the drone that will support the melodic activity more strongly.
I would say that I haven't noticed a particular pattern for which keys employee functional or modal harmony and I'm pretty sure your thoughts are the result of the type of music you listen to. Perhaps try checking out some different genres or widely within the genres you have listened to to see if this changes your thoughts.
