What is the logic to adding notes to chords or stacking thirds ABOVE the initial note?

Accompaniment is generally chord progressions of some sort while melody is generally a line of single notes. The example that led to this discussion is actually a bit odd, because rarely do you see two lines of music maintain major and minor thirds (the result of using alt-3 to add a note) over an extended period of time. The lower line usually varies more to add texture to the song. There is nothing wrong with doing this. I would never question how a master such as Handel wrote his music. The OP didn't say what song he was transcribing, but my experience with Handel is that this is not his norm.

With this in mind, adding a chord to a melody is generally not what one wants to do. Also, the melody often uses notes not included in the chords of the harmony. Melody also tends to have more passing tones than the harmony does, though at times the harmony will follow the melody in rhythm. The melody is very often the highest note as well, since higher notes tend to stand out more than lower notes. Having said all of this, it is not at all unheard of to add chords below the melody. In Piano and Guitar music this is often done through the use of different voices in MuseScore (and other notation software). In larger ensembles, this is usually done by adding the harmony notes to the other instruments depending on how many notes the instruments are capable of playing at once. Since chords are generally based upon the root (lowest note in the chord), it makes more sense to build the chords from the bottom up, no matter how many notes are in the chord. Of course there is an abundance of exceptions to all that I have said, particularly inverted chords, but even these can be called by a different name and often are in music that includes chord progressions.

MuseScore does allow defining intervals below the selected note. I have chosen alt-shift-# to indicate the note is added below in contrast to alt-# for the note to be added above the note. I realize this key combination is not ideal for all keyboards and platforms. Apparently in earlier versions there were a different set of predefined shortcuts that caused problems on some platforms so they were eliminated. My shortuct definitions work well on my Windows 10 - QWERTY keyboard.

As I said in the other thread, you can add note an interval above or below. Check Note > Add Interval and remap the shortcut if you prefer to have shortcuts for note below.

I follow you, as I often transcribe folk music with lot of fifth of the tonality added below the melody.
Such as lot of g if in C major e.g.
Adding them is always [shift] [g] followed by [ctrl] [-]
Not a very big problem, but indeed avoiding all these [ctrl] [-] would be nice.

A few observations that may help clarify things.

First, as to the general question of whether people are more likely to want to add notes above or below: that's like asking if people are more likely to prefer chicken or pork for dinner. Some may almost always prefer one, some may almost always prefer the other, but for many people, it's just going to depend. Yes, there are situations - such as ones in which one is creating new music on the fly - where one is likely to first put down the melody note and then think about adding harmony notes, and those harmony notes are likely in certain context (eg, jazz) to be below the melody. On the other hand, same situation but change jazz to folk, and suddenly harmonies above the melody become more common. Or, change from creating new music on the fly to simply copying an existing piece of sheet music, and suddenly it becomes a lot more common for people to think bottom up, because that's almost invariably how we spell chord. Consider, if I ask you to list the notes of a C major triad, chances are good you'll say "C E G", not "G E C".

So anyhow, there can be no denying that both uses are common and are worth supporting.

Now, one to thinking about why Shift+letter works the way it does. You may or may not recall that 1.3 worked differently: instead of always adding above, it tried to picked the closest pitch, so if you had a C and pressed Shift+E it added above, but Shift+A added below. There were complaints that this felt inconsistent, so a choice was made, and for better or for worse the people involved in that discussion seemed to have a preference for going with adding above.

The story with Alt+number is a bit different. Back in 1.3, Alt+number added an interval above, Shift+number added an interval below. Unfortunately, Shift+number shortcuts created a conflict on French AZERTY keyboards, because numbers already require Shift to access. So the conflict was resolved by undefining the Shift+number shortcuts for interval below, leaving it up to the use to define his own if he wanted.

In hindsight, probably someone should have noticed that this left us with two different ways of adding notes above and none of adding notes below, and eventually someone did, but for whatever reason, it just didn't seem worth changing behaviors that by then people had become accustomed to.

It's all definitely worth revisiting for a future release, but hopefully this provides some insight into why things are how they are.