Every comment above concerns harmony, reflecting the myopic focus of mid-20thc theory. Beethoven's primary influence on the Romantics wasn't in harmony, although, contra Rosen, he had substantial influence there as well. His primary influence was on thematic processes, multimovement structure, and aesthetics.
After Beethoven's essays in the thematic unification of multimovement cycles (Fifth and Ninth Symphonies, Appassionata, some of the late sonatas and quartets) nearly every Romantic composer took up the practice: Schubert, Schumann, Berlioz, Tchaikovsky, Franck, Liszt, Dvorak, Brahms occasionally, Rimsky-Korsakoff and countless minor figures. The practice was even more pervasive in the 20thc, especially among the Russians. After Beethoven, and influenced by his example, the majority of Romantic Era sonatas and symphonies beginning in the minor mode ended in the major mode. All of this reflected a new aesthetic ideal of imbuing large-scale works with a dramatic, quasi-narrative expressive continuity.
Beethoven changed the profile of first movement sonata forms in ways that were highly influential on Chopin and nearly every major Russian and Eastern European composer through Shostakovich.
And it's worth remembering that composers are not only influential for the paths they open, but for those they close or make perilous as well. Someone above mentioned Brahms's reluctance to compete in the symphonic genre, and I imagine many hearing Beethoven's late sonatas and quartets found the prospect of trying to extend or add something to that madness bewildering or daunting. That too is impact and influence on the future.