This answer focuses on answering:
Is there a strong case, grammatically and linguistically, to be made in arguing that the prophecy specifies a long reign instead of being on the throne for a short time? Or is this prophecy somewhat “unfulfilled”?
Notice similar vocabulary but different grammar and meaning of:
וּמֶ֣לֶךְ יִשְׂרָאֵ֡ל וִֽיהֹושָׁפָ֣ט מֶֽלֶךְ־יְהוּדָ֡ה יֹשְׁבִים֩ אִ֨ישׁ עַל־כִּסְאֹ֜ו
(1 Kings 22:10, BHS2003)
is Qal participle masculine plural absolute
The king of Israel and King Jehoshaphat of Judah were seated on their thrones,
(1 Kings 22:10, JPS1985)
The phrase in Jeremiah 36:30 is:
לֹא־יִֽהְיֶה־לֹּ֥ו יֹושֵׁ֖ב עַל־כִּסֵּ֣א דָוִ֑ד
is Qal participle masculine singular absolute.
literally it/there is not to him is a way Hebrew says he does not have, although
is more common, but here expresses a future meaning.
He shall not have any of his line sitting on the throne of David;... (JPS1985)
However, "any of his line" is supplied as understood because the participle is acting as a noun/substantive. It is literally:
He shall not have sitting on the throne of David...
Note how Davidson described this use of the particlple;
Rem. 2. In order to express more distinctly the idea of duration, particularly in past, the verb היה is sometimes used with the ptcp., generally in a clause of circumstance explicative of the main narrative, but also in an independent statement.... Jer. ...; 36:30 ...
Davidson, A. B. (1902). Introductory Hebrew grammar Hebrew syntax (3d ed., p. 136). T&T Clark.
Also note that
is Qal imperfect third person masculine singular expressing a future tense meaning, rather than the perfect
typically expressing a past tense meaning.
Let's look at this in more detail to see what duration means here.
Hebrew does not have past, present, and future tenses. It has perfect (completed action) and imperfect (incomplete action) tenses. Past action is usually expressed with perfect tense and future action imperfect tense. Present action is usually expressed with the participle.
Expressing is and are in Hebrew
While הָיָה is the being verb in Hebrew, pronouns are commonly used for a present tense meaning. The imperfect here has a future tense meaning.
The Hebrew Participle
While the verbal use of the participle in Hebrew often has a present tense meaning, in 1 Kings 22:10
has a past imperfect meaning as it is translated in the Septuagint (LXX). However, in Jeremiah 36:30
is used substantively as a noun. In languages such as Greek and English, a participle wih a being verb forms a paraphrastic, but in Hebrew it tends to make the participle substantive.
So, what does duration mean? If you say he shepherds, it refers to a particular instance. But if you say, he is a shepherd, it says more about what he does. Also, he farms versus he is a farmer. He builds versus he is a builder. Thus, what keeps Jehoiachin from contradicting this prophecy in Jeremiah 36:30 is he does not seem to be recognized internationally as the king of Judah. Nebuchadnezzar carried him off to Babylon and made his uncle king in Babylon.