Context is really important with the subdominant role.
By subdominant, I just mean the IV note, NOT a chord.
I understand what you mean, but melody and harmony are inextricably linked. You can't really separate them. Importantly in tonal music, even if the music is entire a single melodic line, the tonic is a reference point and harmonic relationships are implied. Historically, these tendency tones ideas evolved from counterpoint and harmony.
Question: this doesn't seem to have anything to do with consonance or dissonance (because it is only a single note!)
In the vertical sense, yes, there is only one tone. But music is a temporal art! In time there are other notes. In other words, there are other notes linearly. This is where the sense of stability is created: linear movement to and from the subdominant degree.
Let's call the subdominant tone
FA. How you move to and from
FA is what creates the sensible of stability and resolution.
FA moves down to
MI the harmonic implication is
MI is the mediant and a member of the tonic chord. The critical part is a downward move of a half step is regarded as
MI. In modern harmony we would regard
MI as the upper voice in
V7 I or
viio I. (If
FA were in the bass, it would be
V4/2 I6.) The various harmonic implications are a move to a tonic chord from a dominant harmony.
When the direction is reversed and
MI moves up to
FA the harmonic implication changes.
MI up to
FA can be re-contextualized as
TI up to
DO. In that case, where
FA is treated as
DO the move is to an implied stable chord. In modern harmony it might be something like
I IV regarded as
V I. I think it becomes clear when the tones are in the bass:
I6 IV regarded as
I may not be explaining it clearly. I'm trying to paraphrase an overview from: Gjerdingen, Music in the Galant Style where he provides an "...excursus on eighteenth-century solmization." I had to re-read it about twenty times - and look for it in real score - before I felt like I understood the concept.
The main point is:
Ascending and descending movements to and from the subdominant have different harmonic implications.
Those harmonic implications - even in an unharmonized line - create a sense of stability or in-stability for the subdominant degree.