This is a very old and rather huge debate, so let me try to hit on some important historical context.
Aesthetics, as the modern philosophical discipline we now understand, was developed in the late 18th and early 19th centuries, particularly among a group of German philosophers. At that time, vocal music -- which had specific meaning ascribed through words -- was generally perceived of as the best music. Instrumental music (like symphonies and concertos) was often deemed inferior, as it lacked meaning. See, for example, Immanuel Kant's Critique of Judgment (an important early philosophical treatise on aesthetics), where Kant classifies various types of art into categories about how beautiful they can be, whether they can express "sublime" elements, etc. Kant places instrumental music on par with wallpaper in his classification, which is insightful given the time he was writing: a classical symphony was often something to almost be enjoyed as background music at a palace function, with the orchestra playing while nobles chatted. It wasn't the sort of "high art" where everyone always sat down quietly and in rapt attention to listen as we do at classical concerts today.
Another useful bit of background information is Kant's distinction between two words of ideas -- the phenomenal (accessible to our senses) and the noumenal (the realm of pure ideas, from the Greek word nous which means mind/spirit). The ideal version of knowledge -- and by extension aesthetic appreciation -- was to understand the noumenal world, as philosophers seek to do, to not see the particular physical manifestations of "beauty," but rather to understand "beauty itself" as a higher abstraction.
This is relevant because the view of instrumental music began to shift around the time of Beethoven, when several philosophers began to argue that the very lack of words or specific verbal ideas in Beethoven's instrumental music meant they could tap into an even more sublime and more profound meaning. Today, the most well-known article in that regard is E.T.A. Hoffman's review of Beethoven's Fifth Symphony. As Hoffman writes:
Thus [Beethoven] is a purely romantic composer, and if he has had less
success with vocal music, is this because vocal music excludes the
character of indefinite longing and represents the emotions, which
come from the realm of the infinite, only by the definite affects of
words?
In other words, Beethoven's symphony is "pure" Romanticism, able to portray depths of ideas that tap into the "realm of the infinite" (sort of like Kant's noumenal realm of pure ideas). When Beethoven tries to write vocal music, he doesn't have as much success, as the "definite aspects of words" are just a simple particular manifestation of ideas, not the eternal, "infinite" pure world of aesthetic appreciation created by absolute instrumental music.
Many philosophers and music critics of the early 1800s, particularly in Germanic lands, came to view music in this fashion. One very notable figure in the debate was music critic Eduard Hanslick who wrote a prominent philosophical pamphlet called On the Musically Beautiful in 1854. Hanslick took earlier ideas to a more extreme degree, arguing for what is usually called formalism in music aesthetics -- that is, that the power of music comes purely from the form of the sounds and their interactions, not from any verbal meaning (in vocal music) or any supposed "narrative" (whether in actual vocal text or implied in representation, as in the question's Berlioz example). Famously, Hanslick and Wagner were enemies, because Wagner believed in the power of Leitmotifs to represent specific people, things, places, and ideas in his operas, even when performed only by instruments without text. Hanslick thought that perspective was absurd and that pure "absolute" music, like those Beethoven symphonies, achieved the highest form of musical art by being divorced from text and a narrative meaning.
This argument has gone on ever since, with many composers taking firm stances on one side or the other. For example, Stravinsky notoriously changed his perspective on Rite of Spring -- a work originally composed to go with a specific narrative and accompanied by a rather novel ballet with very rustic choreography (to put it mildly). But later in life, Stravinsky distanced himself from programs and argued that the beauty of Rite of Spring should be best understood in a pure concert form, only instrumental, without the ballet and really not even meant to be a specific representation of the narrative.
The question is right to note that another part of Romanticist aesthetics was about the power of narrative expression. And people like Wagner embraced that to an extreme degree. The formalists, on the other hand, thought that the greatest aesthetic power in music could only be achieved through abstraction.
There are relatively few adherents of the formalist school these days, who would strongly argue that only pure instrumental music without any text or narrative is the highest form of art. But the idea of absolute music has a very long and important place in the history of aesthetics, which is what is being alluded to in the video in the question.
