Where you locate the start of the Romantic period will depend on how you define romanticism. Perhaps the most common musical narrative is that Beethoven (especially starting with his middle period) shrugged off classical norms in favor of a more grandiose, difficult, Heroic aesthetic right about the time that Napoleon was waging the Napoleonic Wars. Such a view of history will see Romanticism as replacing Enlightenment rationalism right about the time of the French Revolution, and will take Beethoven as a sort of "founding father" in the musical realm, and probably Goethe as the main literary figure. I'd say this would likely accord well with how 19th century authors would have constructed their own identity. It's also the narrative most often repeated in standard music history textbooks.

But of course, there are plenty of ways to project the roots of romanticism back into the eighteenth century. Viewing Romanticism as deeply entangled with middle class values and sensibilities, we might point to the rise of that class over the course of the eighteenth century and the emergence of what Habermaus called the "public sphere" in which that class could debate aesthetic and political issues in coffee houses and the like. We could point out the rise of a reading public in England and the flourishing of epistolary novels (that is, a novel written as though it were a series of letters from a real person) like Richardson's Pamela, which purported to give readers access to the private thoughts and emotions of an "everyday person." Such a focus on private emotion was so successful that it created a "cult of sensibility," which we can find echoes of in German "Sturm und Drang" (Storm and Stress) poetry and "empfindsamer Stil" (Sensitive style) music. All of these are mid-century developments that would have preceded and profoundly influenced composers like Haydn and Mozart. This view might take Enlightenment rationalism and an emerging bourgeois Sensibility as two poles that tugged at the literary and musical discourse throughout the second half of the 18th century, the latter sowing the seeds that would blossom into the major aesthetic viewpoints of Romanticism. From this vantage point, someone like CPE Bach would be an important musical figure, and people like Richardson and Schiller would be appropriate literary figures advocating for early romantic sensibilities.