The problem with determining Tiglath Pileser III's origins is lack of evidence.
As Amélie Kuhrt has repeatedly observed in her 2-volume work The Ancient Near East, C. 3000-330 BC, alteration of Assyrian inscriptions in antiquity was commonplace.
In the case of Tiglath Pileser III, we have contradictory evidence. As Dr Floyd Nolen Jones noted:
It is well known and accepted by most Assyriologists that a significant number of the inscriptions claimed by Tiglath-pileser (III) deal with events that precede his reign.
- [Nolen Jones, 2002, p158]
For example, there is a mutilated brick inscription states that Tiglath-pileser
III is the son of Adad-nirari III. However, the Assyrian King List makes him the son of Ashur-nirari V, son of Adad-nirari III [Pritchard, 1969, p566].
This is a problem, since the King List places Adad-nirari III four monarchs before Tiglath-pileser III came to the throne, and depicts Ashur-nirari V as both his father and immediate predecessor upon the throne.
Wikipedia takes a charitable view, and observes simply:
He described himself as a son of Adad-nirari III in his inscriptions, but the accuracy of this claim remains uncertain.
As you mentioned in the question, we know that Tiglath-pileser III was a general named Pulu before he became ruler. We also know that he seems previously to have been the governor of Kalhu/Nimrud [Healey, 2000]. Beyond that, we really know nothing of his origins.
However, the fact that he and his sons took different throne names doesn't actually support the idea that they were foreign usurpers. His assumed name, Tiglath Pileser III, is actually the Hebrew version of the Akkadian Tukulti-Apil-Esara. This was clearly a name chosen to link himself directly to great kings of the past, presumably as a means of legitimising his position.
His son and successor Shalmaneser V was previously known as Ululayu, but this simply means "born in Elul" - Elul being the 6th month of the Assyrian calendar. His throne-name, Shalmaneser, meant "the god Salmanu is foremost", and once again provided him with a link with distinguished predecessors.
Shalmaneser V was succeeded by his brother, Sargon II. We actually don't know what name he was known by before he assumed his throne-name (which meant "the king is true").
As governor of Kalhu/Nimrud, it seems unlikely that Tiglath Pileser III would have been a foreigner. The usual assumption these days seems to be that he was a usurper, probably of royal blood, who took the Assyrian crown by force after engineering a coup against his ineffective predecessor. There is no suggestion, however, that he was a foreign usurper.
However, as I said right at the start, we really don't have the evidence to be at all certain.
That said, there are probably many theories about Tiglath-pileser III's origins, but none, so far as I am aware, that are ascribed to by reputable Assyriologists.
Healy, Mark: The Ancient Assyrians, Bloomsbury, 2000
Kuhrt, Amélie: The Ancient Near East, C. 3000-330 BC (2 vols), Routledge, 1995
Nolen Jones, Floyd: Chronology of the Old Testament: A Return to the Basics, Kingsword, 2002
Pritchard, James, B: Ancient Near Eastern Texts Relating to the Old Testament, Princeton University Press, 1969