In a minor key, the chord built on the tonic can be a minor sixth or a minor major seventh chord, both of which are supposed to have tonic function. To my ear, the minor sixth chord can feel like "home", but I've always heard the minor major seventh chord as being too unstable to sit on because of the augmented triad inside it. How is it supposed to have tonic function then? (I know it's the I chord in harmonic (or melodic) minor. My question is how it can have its supposed harmonic function.)
While generally in functional harmony chords build on the i are considered to have tonic function, this is not always the case, especially when using non-traditional chord like the minor 6th or the minor major 7th. The minor 6th does not come from either the natural or harmonic minor scales, the scales most often used to build chords in the minor key. It can be considered to come from the melodic minor scale as you state in your question. I have mostly heard that it's from the dorian mode. On the other hand, the minor major 7 can be considered to be build using the harmonic minor scale. This is an atypical use of the harmonic minor scale as mostly it is just used to build a dominant V chord (or sometimes a viio chord). To my knowledge, both of these chords come from jazz theory and do not follow the typical rules of functional harmony.
So, to the nitty-gritty. Both minor 6ths and minor major 7ths can be considered to have tonic function, but do not necessarily have tonic function given certain circumstances. The tritone in the minor 6th and augmented 5th in the minor major 7th are prone to sounding resolved when moved. A minor 6th build on the tonic is enharmonically equivalent to a diminished 7th built on the 6th scale degree. Meaning it can act as a secondary dominant, specifially the viio/♭VII, iio/v, ivo/III, or vio/♯i (not 100% sure about the last two). Further, I've seen the minor major 7th resolve to the ♭III chord; the augmented 5th interval falls down to a perfect 5th interval. This can tonicize the relative major. All that is to say, these chords can be very directional and have clear dominant function even if build on the 1st scale degree.
If you want to establish tonic function with one of those chords here are a few suggestions:
- Stay on the home chord. Eventually, any chord or structure you sit on will become tonicized.
- Try a two-chord vamp. Switch between your chord of choice and the root triad.
- Move to the minor 6th or minor major 7th chord with a dominant function chord like the V or the viio.
- Move away from minor 6th or minor major 7th chord in such a way that doesn't show a clear resolution. See above for chords not to go to.
Lastly, to your point of minor major 7th chords sounding less able to be tonicized than a minor 6th chord. Quite possibly this is due to the fact that we are much less familiar with augmented triads than diminished ones in western music. Particularly, this is true if you listen to a lot of blues where dominant 7th chords are ubiquitous (the upper structure being a diminished triad).