The etymology seems the same and according to sources online, the Arabic transliteration of 'Hazarmaveth' is actually 'Hadramawt'.
What is the academic interpretation of this similarity?
Did what is now Yemen have much importance during the time of the Old Testament? It appears as if Ancient South Arabia had a strong influence on the Levant and vice versa, is this correct?
Short answer: yes, it is most likely a Hebrew approximation of the toponym <ḥḍrm[w]t>.
Longer answer: The Biblical context almost certainly seems to point to a pre-Islamic South Arabia, as both Hazarmavet (חֲצַרְמָוֶת) and Sheba (שְׁבָא) are mentioned side-by-side, once in Genesis and later in Chronicles 1. Perhaps a Biblical scholar can correct me on this, but the writing of Genesis is usually dated to c. the 6th to 5th century BCE, at which point both the kingdoms of Sabaʾ (s¹bʾ) and Ḥaḍramawt (ḥḍrm[w]t) were active as political players and mercantile contacts between the Levant and South Arabia were extant.
On philological grounds, it seems likely that the Saba' we know from pre-Islamic South Arabia and Biblical Sheba are the same. The normal outcome of the Semitic sibilant s¹ is Hebrew šin, so that makes sense. I am personally more critical of the connection between what appears to be the legendary Queen of Sheba and pre-Islamic Saba, but that's a different subject.
Historically, this connection is also plausible. Genesis mentions a number of other toponyms, including the oasis town of Dadan, so it makes sense to imagine that these names reached ancient Israel through trade. Whereas Sabaʾ was culturally and politically the most influential of the South Arabian states, Ḥaḍramawt was one of the few places where frankincense could be harvested. It is not unlikely that it is due Ḥaḍramawt's mercantile importance that the name found its way into Genesis.
On the etymology of Ḥaḍramawt: the origins of Ḥaḍramawt as you might find on Wikipedia i.e., ḥaḍar(a) mawt is very cool and Lovecraftian, but almost certainly wrong. It seems more likely that the ending -ūt is similar to that found both in Levantine toponyms (e.g., Bayrūt, Qaryūt) as well in Modern South Arabian (e.g., Raysūt, Ḥabrūt, etc.). Further establishing an etymology is problematic.
Sources
Fedele, F. 2014. Camels, donkeys and caravan trade: an emerging context from Baraqish, ancient Yathill (Wadi al-Jawf, Yemen).
Rahkonen, P. (2015). Some Semitic Toponymic Types
Robin, C. 2001. Sheba II (Supplément au dictionnaire de la Bible)
Zadok, R. (1997). A Preliminary Analysis of Ancient Survivals in Modern Palestinian Toponymy