I would say that in general there is no good reason to expect the first note of a melody to be equal to the finalis of its mode. It happens, but it also doesn't happen. And the same for the psalm tones: in the Liber the intonation is not the final for half of the psalm tones.

Of course there are alternate endings for most of the psalm tones, and those helpfully often end on the pitch the antiphon starts on. But not always...

Regarding the Psalm tones: They are made up of an intonation, a reciting pitch, and a mediant and final cadence.

The intonation has nothing to do with the final of the mode, it is simply an approach to the reciting tone/Tenor/dominant, historically from a 3rd below. (Long, convoluted answer below.)
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In the Authentic Modes: 1, 3, 5, 7, the reciting tone is on the fifth degree of the modal scale: so in Mode 1 on D (re), the reciting tone or Tenor is on A (la); in Mode 5 on F (fa), the Tenor is on C (ut); in Mode 7 on G (sol), the Tenor is on d (re). In Mode 3 on E (mi), the Tenor was originally on B (si), but gradually shifted up to C, a more stable note.

In the Plagal Modes: 2, 4, 6, 8, the Tenor is on the third degree of the scale: so, in Mode 2 on D (re), the Tenor is on F (fa); in Mode 4 on E (mi) the Tenor is on G (sol)--though this also shifted up); in Mode 6 on F (fa), the Tenor is on A (la); and in Mode 8 on G (sol), the Tenor is (or was) on B (si)--this too shifted up.

With this in mind: The intonation of the psalm tone, in All modes, Authentic and Plagal, was originally simply an approach to the Tenor from a third below it. So, in Mode 1, the intonation is F (fa) G (sol) A (la); in Mode 2, D (re) E (mi) F (fa); in Mode 3, G (sol) A (la) B (si); in Mode 4, E (mi) F (fa) G (sol); in Mode 5, A (la) B (si) C (ut); etc.

Over the centuries, particularly with the introduction of Polyphony, some pitch changes took place to avoid certain weak pitches, notably B (si), but also E (mi). So you have the more modern Tenor of Mode 3, migrating up to C from B, which causes the intonation to change form G-A-B, to G-A-C, the same for Mode 8; in Mode 2, the intonation migrated down to avoid the E (mi), and so changed from D-E-F, to C-D-F; in Mode 4 the intonation changed in two ways historically: from E-F-G to D-F-G, then to G-F-G; this was later modified again with the raising of the Tenor from G to A, and the intonation became A-G-A.; the intonation of Mode 7: B-C-D, was ornamented to begin on a strong pitch rather than a weak one: C-B-C-D; Etc. Some of these variants can be seen in the Toni Psalmorum in the 1934 Antiphonale Monasticum. Of interest is the Tonus Solemnis of Mode 5, which has the same intonation as Mode 8, an ornamented form of G-A-C (and Mode 2, transposed, C-D-F), which shows that its intonation dropped through the centuries.

PDF of the section of the Antiphonale Monasticum 1934, with the relevant section here: http://archive.ccwatershed.org/media/pdfs/13/12/10/12-07-56_0.pdf, the Toni Psalmorum begin on Pg. 1210 of the book; which is pg. 225 of the PDF.

Sometimes the choice between the G and A Tenors has to do with the antiphon that it is attached to. For example: Looking at Fr. Weber's Communion for the 23rd Sunday: "Like the deer", the melody of the first two settings, but especially the simpler setting ii, sits on or around G quite a bit, which necessitates the use of the G Tenor in the psalm-tone, otherwise it would sound strange to be reciting on a G for the Antiphon and then suddenly recite on an A in the Verses. (Similarly, when I sing his setting of the Communion for Ash Wednesday "He who ponders", Mode III, which is also one of the communion antiphons in the Common of Doctors, I change the reciting tone of the Psalm verses to B, because that in the Dominant that is used in the antiphon.) Also, it should be noted that the G recitation in Mode IV, while part of the most ancient practice, completely fell out of use centuries before Solesmes's revival of Gregorian chant ca. 1900; e.g., I have never seen a single Falsobordone (polyphonic psalm-tone) or organ verset/intonazione/toccata that utilizes the G tenor, all use A.

As far as your schola's difficulty with 4g: I can't say. I can only guess that it is perhaps a question of familiarity. In a quick survey of Weber's Proper of the Mass for Sundays and Solemnities:

There are 9 Mode IV Introits, all of which use the Solemn Introit Tone for the verses, which recites on A.

There are 14 Mode IV Offertories, only two of which use the G tenor for the Verses, and the majority of those with the A tenor use the Solemn Introit Tone for the verses.

There are 12 Mode IV Communions, half of which use the G tenor, the other half use A (though half of those are in the transposed system, with the tenor on D and the final on A, but melodically the psalm-tone is the same as the A recitation). And, of those with G recitations, one of them is specifically connected to the simpler setting, ii, of the Antiphon, the first setting (i), uses A.

There is one 'Varia', the antiphon "Blot out my transgressions" to Psalm 50(51) at the distribution of the ashes on Ash Wednesday, which uses the G tenor for the verses.

There are, then, in total 36 Mode IV chants, only 9 of which (1/4) use the G tenor for the verses (2 Off., 6 Comm., 1 Varia).

I think the OP was about the last note of the antiphon not matching the first note of the intonation of the psalm tone, and of course it all depends on which antiphon is being used. In general, the antiphon kindly leads to the psalm-tone intonation, but it isn't required to do so. I sometimes think the variations on the psalm-tone endings (the pitches for E U O U A E) are to help at the other end, so the ending lands near the first pitch of the antiphon. One finds kindnesses and challenges depending on the antiphon.