Bhante @Sujato has previously examined the etymology of the term:
I was able to find two relevant EBTs in which we see the Buddha explaining why he would use the title tathāgata:
"Concerning things past, future and present the Tathāgata is a prophet of the hour, a prophet of fact, a prophet of good, a prophet of the Norm, a prophet of the Discipline.
For this is he called Tathāgata.
Whatever, O Cunda, in this world with its devas and Māras and Brahmās, is by the folk thereof, gods or men, recluses or brahmins, seen, heard, felt, discerned, accomplished, striven for, or devised in mind,—all is understood by the Tathāgata.
For this is he called Tathāgata.
And all that in the interval between the night, O Cunda, wherein the Tathāgata was enlightened in the supreme enlightenment, and the night wherein he passed away without any condition of rebirth remaining,—all that, in that interval, he speaks in discourse or conversation or exposition:—all that is so, and not otherwise.
For that is he called Tathāgata.
As the Tathāgata says, O Cunda so he does; as he does, so he says.
Inasmuch as he goeth even according to his word, and his word is according to his going,
for that is he called Tathāgata.
As to the world, O Cunda, with its Māras and its Brahmās, of all its folk, divine or human, recluses or brahmins, the Tathāgata hath surpassed them, hath not by them been surpassed, surveys them with sure vision, disposer of things.
For that is he called Tathāgata.
–DN29
and
"Whatever is seen, heard, sensed, cognized, reached, sought after, examined by the mind—all that the Tathāgata has fully awakened to;
therefore he is called the Tathāgata.
Whatever the Tathāgata speaks, utters, or expounds in the interval between the night when he awakens to the unsurpassed perfect enlightenment and the night when he attains final nibbāna, all that is just so and not otherwise;
therefore he is called the Tathāgata.
As the Tathāgata speaks, so he does; as he does, so he speaks. Since he does as he speaks and speaks as he does,
therefore he is called the Tathāgata.
In this world with its devas, Māra, and Brahmā, among this population with its ascetics and brahmins, its devas and humans, the Tathāgata is the vanquisher, the unvanquished, the universal seer, the wielder of mastery;
therefore he is called the Tathāgata.”
– AN4.23
P.S.: @Yasoja kindly pointed me to Iti112.