All the chords are the same with low G linear tuning. Re-entrant tuning gives a uke, especially the soprano it’s unique voice.
Me being no expert, but having an opinion....
I think the term comes from the fact that ukes have always been strummed in an up & down manner, unlike other instruments which are normally strummed downwards, (as guitars used to be).
I tend to down strum when playing ukes, more so than down up, but then I use linear low G mainly, (or to be more precise, almost exclusively).
Since "re-entrant" refers to the breaking (or really, restarting) of a rising or falling pitch pattern across the fretboard, re-entrant tunings really are re-entrant. If you reordered the strings CEGA (all in the same octave), the tuning would be linear, not re-entrant. Linear doesn't refer explicitly to having G an octave lower; it's just that if the G is an octave lower following the GCEA pattern, the result is a linear tuning: the pitches consistently rise as you cross the fretboard. If you drop both the G and A an octave (as some do), you have another type of re-entrant pattern: rising until you reach the A string, where the pitch falls.
It is not true that the roots of most chords fall on the third string (in re-entrant tuning); I think you mean that the lowest pitch typically falls on the third string, but this need not be the root. We freely use inversions on ukuleles—in fact, more often than we use "root position" (i.e., uninverted) chords.
The first string isn't always redundant in major triad chord shapes. A triad by definition will have only three distinct pitches, so if we strum or pluck all four strings, one string must double the pitch of another, either in unison or at an octave. And while the 1st string often doubles the pitch of the 4th string in triads, it's also common for the 1st string to double the pitch of the 3rd string, an octave higher, or for the 2nd string to double the pitch of the 4th string, an octave higher, or for the 2nd string to double the pitch of the 3rd string, in unison. In other words, in some common triad chord shapes, the 1st string does not double another pitch but plays that triad component all by itself (and is thus essential). Nor in other chords is it always an added note to the triad, like a 6th, 7th or 9th; more often than not, it plays one of the core triad pitches, undoubled, with color being added on one or more other strings.
One reason that the fleas pattern (whether re-entrant or linear) works so well for chording is that the open pitches divide the octave about evenly, unlike with most other tunings, like all-fourths, all-fifths and "open" (triad) tunings. As for why the strings can't simply be ordered lowest to highest (CEGA), it just turns out that other orderings produce less friendly fingerings for most chords, especially when you consider all keys and all chord types.
For me the reentrant part of Uke tuning has always inferred that the 4th string should be a G below the C and that a g above the C is used as a mater of necessity. I feel that that inference is misleading and that your last sentence captures why gCEA is as it is.
A question: Are instruments like the tiple (gG cCc eEe AA) or the 12-string guitar considered reentrant?