I'm going to talk about Germany a lot. Yes, I know, Vienna is in Austria. German speakers however, decided that they really liked a lot of this music and because of shared language that it was "theirs". Generalizations do occur, especially when talking about France. Beethoven's symphonies arrived in France pretty late compared to the rest of Europe, and were not that well received, as there seems to be historically some "beef" from France towards German composers. This answer honestly cuts a few corners. German music and musical pedagogy did disseminate to the United States a lot earlier than Arnold Schoenberg and other composers that were labeled "degenerate" by the Nazis. We know that Beethoven was certainly in the ears of American Composers before the 20th century: Charles Ives quotes a lot of Beethoven, Schubert, Bach, among others, in his music. He studied with an American, so we can be assured that Beethoven got to America before his generation.

To give a short and simple answer before my long answer: "nationalism", touring virtuosi, transcriptions, and the foundation of Music Academia.

Around the time where Mozart, Haydn, and Beethoven were all getting jiggy with it in Vienna, there was also a movement for the revival of "old" music, particularly in a way that emphasized a Germano-centric pedagogy, centered around the works of JS.Bach and his contemporaries (with Bach in particular being "on the throne"). Beethoven, Schubert, Mozart, and Mendelssohn in particular were caught up in the hype. Coincidentally, this is just when the German music printing industry had started to take off. This was coupled with German journals of music criticism coming into existence, reviewing the works of Beethoven, Schubert, and Bach. This storm of interest in earlier german music, creation of a German musical identity, and the printing industry meant the perfect storm for the beginnings of a pedagogy based on German music.

Composers throughout the 19th century engaged with the German tradition (except the French, who really hated it). The theme itself is a fantastic example of a common compositional strategy called "thematic transformation", or in a really fancy German term thematische Verwandlung. This technique is essentially stretching and pressing the notes of the theme in various ways. Beethoven 5 is chock full of this sort of examples of this (augmentation, diminution of rhythm). Composition pedagogy in the so-called German school consisted of understanding and grasping the concepts of thematische Verwandlung, knowing and familiarizing one's self with the works of Bach (as a means of learning counterpoint).

While all of this was happening, we had a few people going around Europe disseminating the works of Bach, and the works of Beethoven. For example, Franz Liszt (incredibly popular touring musician. The so-called "first superstar") arranged a Bach Organ Prelude and Fugue for piano, as well as, you guessed it: DUH DUH DUH DUUUUUH . Clara Schumann, and her husband also went around performing arrangements of Bach, as well as Mendelssohn. Why is this important? Because it is a lot easier to introduce people to new music if it takes less money to show them the music. Orchestras cost a lot more money than one pianist So people were starting to become familiar with the music of Bach and Beethoven through famous touring virtuosi (this includes England). France was still notably not into this whole tradition. They don't particularly matter to this discussion, but I should mention the because the French and German schools seem to have had some sort of aesthetic dislike of each other.

This German composing tradition continued on through the 19th century to Wagner (who was super into German nationalism), Brahms, and Mahler. Then on to our good friend Arnold Schoenberg. Schoenberg was a pretty big deal, you might know him for his innovation in 12-tone serialism. Well, Schoenberg and several cronies all fled to the United States, and with them brought German pedagogy with them just at a time when Music was trying to find its place in the Academy. Thus, music majors studied a lot of Beethoven, particularly if they wanted to be composers. Many institutions practiced the German pedagogical tradition that Schoenberg brought over (and also practiced serialism) because it was seen as the "future of music". Tonality was out, but Schoenberg and his germano-centric 12-tone serialism was in, as were the academic resources he used to teach.

Schoenberg's book "Fundamentals of Music Composition", uses entirely examples from the "German" tradition. And it's mostly Beethoven.

So essentially, why a lot of people know Beethoven 5 is due to the music people studied, good timing on the printing industry's behalf, and fortunate dissemination founded by a creation of a nationalist musical identity. Beethoven is sorta on a pedestal constructed by the tradition that perpetuates his music.

Gonna add some sources that I found useful:

These two books talk about Beethoven as a composition model for various composers. Ive's essays highlight to the reader how spiritually important Beethoven was to his process and conception of music.

Block, Geoffrey. Ives: Concord Sonata. Cambridge: Cambridge University Press, 1996.

—. Schubert's Reputation from His Time To Ours. Hillsdale: Pendragon Press, 2017.

Gingerich, John M. Schubert's Beethoven Project. Cambridge: Cambridge University Press, 2014.

Ives, Charles. Essays Before a Sonata and Other Writings. New YYork: W.W. Norton and Company. Inc., 1962.

Here are some books and sources that talk about about Bach reception and the dissemination of his music, as well as the beginnings of nationalist pedagogy and musical identity:

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Mintz, Donald. “Some Aspects of the Revival of Bach.” The Musical Quarterly, vol. 40, no. 2, 1954, pp. 201–221. www.jstor.org/stable/739671.

Stinson, Russell. The Reception of Bach’s Organ Works from Mendelssohn to Brahms. Oxford University Press. Oxford. 2006.

Warner, M. A. (2013). A Little Prelude and the Scherzo That Almost Wasn't: Beethoven's “New Path” and the Early-Nineteenth-Century Bach Revival. Journal Of Musicological Research, 32(2/3), 83-115.

Wolff, Christoph. "On the Recognition of Bach and 'the Bach Chorale': Eighteenth-Century Perspectives" in Bach: Essays on His Life and Music. Harvard University Press, Cambridge. 1991. 383-390.