Feynman is being... liberally creative. What he says is his own interpolation that "makes sense" from the perch of today. "Must have been psychologically wonderful", perhaps, but "freeing of man from the intimidation of the ancients" is not how the men of Renaissance generally felt. The intimidation they sought the freeing from was not of the ancients, but of religious and scholastic orthodoxy, to which "the wisdom of the ancients" was something of a cultural ground and counter. The feelings toward the ancients were more along the lines of awe and admiration. Ancient works were avidly sought after, translated, transmitted, and enthusiastically built upon. The Renaissance does translate as re-birth, it is not liberation from the ancients that it championed, but their revival, and validation by them.
As for the specifics. First, the discovery of the solution to the cubic was made 30 years before Tartaglia, by del Ferro, and Tartaglia knew about it. Del Ferro did not make it public, but entrusted the secret to his son-in-law and successor Fiore. Tartaglia rediscovered it independently, but knowing that a solution exists is a powerful incentive for looking for it, a nerve-recking experience as we know from Tartaglia's own description, and not giving up. In fact, he rediscovered it while preparing for a "mathematical duel", a common practice at the time, he was called to by Fiore, see Why is “Cardano's Formula” (wrongly) attributed to him? and Saiber's Niccolò Tartaglia's poetic solution to the cubic equation for more on the story.
Second, it is not clear that the men of Renaissance believed that the methods they discovered were unknown to the ancients, I am not aware of Tartaglia, or any of his contemporaries, expressing a Feynman-like sentiment. Instead, there was at the time a popular "conspiracy theory" that the ancients deliberately hid some of their "secrets of the art". Around the time of Tartaglia, Copernicus, looked for affirmation of his heliocentric system in the writings of ancient Pythagoreans, like Philolaus and Ecphantus, and his contemporaries saw it (mistakenly) as a revival of the Pythagorean system. In the preface to De Revolutionibus, he also writes of the reluctance of the Pythagoreans to share their discoveries with the unlearned commoners, see SEP, The Renaissance: Ficino, Pico, Reuchlin, Copernicus and Kepler. Even later, when it came to the calculus methods, Torricelli wrote:
"I would not dare to affirm that this new Geometry of indivisibles is a totally new discovery. I could more easily believe that the ancient Geometers have made use of this method of discovery of the most difficult theorems, though they preferred another way in demonstrations, either to conceal the secret of the art, or not to give any occasion of contradiction to their jealous detractors".
Wallis and others expressed similar opinions. After quoting them in Cavalieri and Euclid, see Revolution and Continuity, p. 159, de Gandt further comments:"By claiming that the ancients possessed analogous instruments, the creators felt justified in adopting new ways. Their invention was a restitution of the ancient skill...". Solution to the cubic was, no doubt, inspiring, but not in the way Feynman imagined.