In order to answer your question, the question itself needs to be modified. To correct your thought, the Romantic Period did not occur specifically during Beethoven's lifetime, so it could therefore not have happened during his "middle" period.
It is important to understand that when talking about labeling a period of music is to label a zeitgeist of thought, and because of that, it is fairly impossible to distinguish a particular moment or instance in which the shift occurred. That said, with every musical period there are events which reinforce the current zeitgeist. This is consistent throughout history. These events are related to the people that we end up studying in school; such as Beethoven.
Beethoven himself is seen as one of the strongest (and the most famous) connections between the Classical period and the Romantic period. However, it is generally considered that for music, what would become the Romantic period began to seriously gain momentum in the early 1830's, a few years after Beethoven's death in 1827.
It is because of this that there cannot be scholarly consensus on a single piece of music that would influence and create and entirely new musical period. At best, a single piece like that would have been seen as an intriguing outlier and imitated by less famous composers.
Regarding the technical explanation that you have heard, that is just not true. Long before Beethoven's time, Mozart and other composers were using 13th chords, non-functional harmony, chromatic passing tones, and non-local harmonic modulations. Beethoven would often use unresolved, harmonically non-functional fully-diminished chords as a means of sustain tension throughout a passage. In comparison to a dominant 9th chord, it's like whipped cream.
In addition to expanded harmony, the Romantic period was a result of composers playing with conventions - both with formal organization, with listeners' expectations, and the beginning of the composer as a self-developing artist.
At the risk of turning this into a paper, I think I'll stop there.
I hope that answers your question.