: Consonance vs Stability In this source, the guitar teacher claims that the m6 chord is more dissonant but also more stable than the m7 chord. He also says that consonance is not always correlated with stability. I can feel

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If not consonance, what actually makes a chord be stable?
Consonance does make a chord stable, but stability and instability are relative. A chord isn't so much stable or unstable as it is more stable or less stable. So, considering chords that all share the same root, and played individually (rather than as part of a progression), a major triad is highly stable, a diminished triad is highly unstable, and m6 is less stable that a major triad but more stable than m7, which is more stable than the diminished triad.
However, context matters. If you have a piece predominantly in C major, and you suddenly toss in a D major triad, that triad will sound far less stable than if the piece had been in D major or G major. The same holds true for any chord. The more alien it is from the primary (local) key area, the more unstable its sound.

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There might contention that sixth chords do not exist and are just a 1st inversion of a minor 7th chord. I was taught this way and it's taken years to accept the sixth chord as being legitimate! In common practice harmony (c. 1600–1900) a sixth chord does not exist, but in harmony after it does.
(There's also a sixth chord in common practice harmony which is different i.e. the 1st inversion of a chord [Wikipedia: Sixth chord explains further], that's not how I'll refer to a sixth chord from here forward.)
Sixth chords are root chords: they function as a root chord.
Minor 7th 1st inversion is not a root chord and are less stable than a root chord. Root chords are happy staying where they are (if their 5th is a perfect 5th) but can move; 1st inversion chords (or other inversions) need somewhere to move to, to resolve.
Sixth chords are often I chords, so a chord progression with sixth chords on the I naturally grounds the sixth chord too, to being even more stable.
Your example, you have a minor sixth chord, as a root chord.
It occurs on:
i, ii, iv, -vi (enharmonically)
degrees of the melodic minor scale.
The corresponding half-diminshed chords (also called minor 7th flat 5) occur on:
+vi, vii, ii, vi (again enharmonically).
Diminished chords in general (and the half-diminished also) are very unstable and want to move. And in changing key, they are, maybe the most flexible in what chord they can move to (they have many options of chords that they can resolve to that sound good).
So I'd summarise that the half-diminished chord is more begging to move than the sixth chord which is quite happy where it is.

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what makes a chord feel "resolutive" ?

It is our expectation of consonance and stability. The leading tones in a progression
are provoking the dissonance and evoking the tension for resolving.

if not consonance, what actually makes a chord be stable?

It‘s actually the consonance! But in Jazz a major 6 or maj 7 chord are considered as stable (at least as final chords).
In traditional western music the sixth ajouté was considered as a consonant chord like the dominant V7 (even the latter has a great tension to resolve the tritone fa ti => to mi do.
So it depends of the function and the chord progression. As the V - I is dominant functionof the tonic this dominant chord so ti re wants to resolve to do mi so (g b d to c e g).
Even that both chords are major triads it is in functional listening and thinking logical to say: the triad of the dominant chord is less stable than the tonic.
While a tonic chord e.g. C maj7 is less stable than C6. The ear that isn‘t schooled listening to Jazz is expecting that the 6th will resolve to the 5th and the 7th to the octave. Now the tension of the 7th is greater than the 6th because of ti as lead tone and the minor second is more dissonant.
I hope you understand that there is a correlation between consonance and stability and that it is depending additional of our listening habits.
Edit:
My explanations above are referring to major chords.
Your question is concerning minor chords with added 6th and 7.
In this case I‘d leave the theoretical background of functional theory and dissonances, I can only refer to my own personal listening habits: as final chord I‘d hear am6 as more stable than am7, as passing chords they are both similar unstable to me.