Of course traditionally Theravada Buddhism has regarded Nagarjuna as not one of its own. He has too long been claimed by Mahayana to expect otherwise. But some relatively recent scholarship regards him as a transitional figure, with close affinities -- as you suspect -- to Early Buddhism. In fact, a very strong case can be made that he regarded himself as a defender of Gautama as recorded in the suttas. His opponents were not in the suttas, but in the Abhidharma philosophy that dates from several centuries after Gautama and several centuries before Mahayana.
A.K. Warder's rather monumental Indian Buddhism argues that Nagarjuna was closer to Gautama than the Mahayana, and blurs the distinction.
David Kalupauhana's Nagarjuna: Philosophy of the Middleway is an eminently scholarly translation and commentary on Nagarjuna's Mulamadhyamakakarika which argues that Nagarjuna was not Mahayanist at all, and in fact makes the case that the Karika itself is a commentary on the Kaccayanagotta Sutta, upholding Gautama's view of the middle way between "eternalism" and "nihilism."
You can find Velez de Cea, “Emptiness in the Pali Suttas and the question of Nagarjuna's orthodoxy.” (Philosophy East and West 55(4) October 2005) on line. Here, parallels between Nagarjuna and the suttas are gathered, and Nagarjuna's roots in the suttas is demonstrated.
In the Sunna Sutta, Gautama is asked: “In what respect is it said that the world is empty?" and answers; "Insofar as it is empty of a self or of anything pertaining to a self.” This is pretty much Nagarjuna's view. In the Sunyatatasaptati (Emptiness: The Seventy Stanzas), he writes that “since the intrinsic being [self-nature] of all entities does not exist . . . therefore they are empty."
The reason I have this at my fingertips is because I've been spending my time in COVID lock-down writing a piece on Nagarjuna that argues that he believes he is upholding Gautama's Dharma. Although it's dressed up, it was really written just to pass the time, but I'll put it on line for you, and in fact would be pleased if you read it, and comment if you wish. It will be at
Reading Nagarjuna -- A rendition of Chapters 1, 24, and 25 of The Mulamadhyamakakarika for Students of the Buddha's Dharma
You are right that Nagarjuna and Gautama both reject the unanswerable questions" about Ultimate Realities (and N. warns that only fools confuse emptiness with the Absolute), believing, in distinction to almost every religion, that answers to metaphysical questions are not necessary for "salvation." As I suspect you know, the Cula-Malunkyovada Sutta asks "did I ever say to you . . . live the holy life under me, and I will declare to you that 'The cosmos is eternal,' or 'The cosmos is not eternal . . . [etc.]. The final chapter of the Karika, “Examination of Views,” is concerned entirely with the unanswered questions contained in this and other suttas. The last verse of the Karika states: “I reverently bow down to Gautama who, out of compassion, has taught the true doctrine in order to relinquish all views.” The "views" he refers to are the unanswered questions of the suttas. No more no less.
I think it is clear from the suttas and Karika that there are 2 reasons for rejecting the unanswered questions -lst because they are improper questions in the sense that they have no answers, they are metaphysical speculations 2nd because they do help --- and N. echoes G. on both counts.
You can certainly, IMO, like me, be a follower of the Buddhism of the suttas and accept almost all of N.'s Karika. But whether a strict Theravedan would agree, I cannot say.