First off, I'm going to assume that when you say "Classical Music" you mean it in terms of the common definition, to mean orchestral style music, opera and the like, so classical as a genre, instead of a time period. (To clarify, Classical is also a time period, generally about 1750-1825, so Mozart would be a Classical era composer, whereas Bach would be a Baroque era composer, who people today would refer to as classical, because he wrote in the genre which we now consider as "Classical")

If that is your meaning, you're going to have a harder time with this question, as what you would likely consider Classical genre music, would include the vast majority of all music being composed from 1700-1900. Even music like La Marseilleise would be written in that style.

But yes, there is a great deal of revolutionary music being written by all sorts of composers during this time. Consequently, there are also a lot of composers being commissioned by nobility or government to help bolster the state.

Take Beethoven, for example. He wrote his 3rd Symphony, now called the Eroica Symphony and dedicated it to Napoleon, believing him the hero of his revolutionary ideals. He then, supposedly removed the dedication when he discovered the horrors that Napoleon was inflicting on the populus, and was furious. Ferdinand Ries, Beethoven's secretary recorded the incident in his memoirs:

In writing this symphony Beethoven had been thinking of Buonaparte, but Buonaparte while he was First Consul. At that time Beethoven had the highest esteem for him and compared him to the greatest consuls of ancient Rome. Not only I, but many of Beethoven's closer friends, saw this symphony on his table, beautifully copied in manuscript, with the word "Buonaparte" inscribed at the very top of the title-page and "Luigi van Beethoven" at the very bottom. ...I was the first to tell him the news that Buonaparte had declared himself Emperor, whereupon he broke into a rage and exclaimed, "So he is no more than a common mortal! Now, too, he will tread under foot all the rights of man, indulge only his ambition; now he will think himself superior to all men, become a tyrant!" Beethoven went to the table, seized the top of the title-page, tore it in half and threw it on the floor. The page had to be re-copied and it was only now that the symphony received the title "Sinfonia eroica."

There's some debate about the veracity of Reis memoirs, as he was known to embelish or forge some of Beethoven's documents, in the spirit of full disclosure, but the story seems quite likely.

Now, would this symphony, even if it had been truly revolutionary, sparked any changes in the general populace? Probably not, as orchestral music was not very accessible to the lower classes, if at all.

As for your question, I need to address the idea of "popular accessible culture". With the exception of the church, orchestral-style music, even very small chamber works were not accessible culture. They were very much the vestiges of the upper class. Think of it this way. Imagine if you wanted to listen to a piece of music today, and the only way to do it was to hire a group of musicians to play it for you. So every time you want to listen to a Zeppelin tune, you have to hire Led Zeppelin, or a decent cover band, and pay all the musicians for their time, travel, expenses, and so on. If you were rich enough, you'd just keep this band on retainer, and trot them out to play for you when you wanted some music, or had friends over, or had a party. So yeah, music was NOT "popular accessible culture" except for the people with money.

The exception to this, somewhat, was the church. While the nobility had the money to pay to hire a group of musicians, so did the church. J.S. Bach is a good example of this. He was hired by, and worked for decades for St. Thomas Church in Leipzig. A significant chunk of his work are what are called Cantatas, which is like a small symphony with musicians, using the theme and text of that particular day. And not just Christmas, or Easter, but the 2nd Sunday of Advent, or the 3rd Sunday of Lent, the 5th Sunday of Epiphany, and so on. And Bach wrote so many that you could attend his church for 7-9 years straight without ever repeating a piece of music.

So by virtue of being a church, that kind of music would be far more accessible than what was being written by the non-church composers, who wrote almost exclusively for nobility, or those with money.

So many composers were influenced (positively or negatively) by revolutionary ideals. Debussy, Shotakovich, Prokofiev, all wrote during times of great upheaval, and it had a huge influence on their work. But if you asked a French peasant to sing you some Debussy, it would be pretty unlikely if he could. So to kind of answer your question in a left-handed way, yes, music was influenced by the revolution(s), but I don't know if any of the revolutions were influenced greatly by music.

Ries, Ferdinand. Beethoven Remembered: The Biographical Notes of Franz Wegeler and Ferdinand Ries (translated from the German.) Arlington, VA: Great Ocean Publishers.

The Classical Era of Classical Music starts with the death of J. S. Bach, the great composer of fugues. During this time period there is a change in how music is presented. Rather than small chamber works focusing on ornate compositions, the works got larger and grander for a formal audience (the exception being opera). Works were written by composers either employed by a family (such as Joseph Haydn) or comissioned by them (such as with Mozart and Beethoven). This this formal music doesn't represent the larger ideals of philosophy (Beethoven does but he's an odd exception) but rather the interest of the ruling class that gets swept away by the French Revolution.

The Enlightenment isn't exactly tied to the musical scene of the era. The Enlightenment, concerned with politics, metaphysics, and society does not liberate or change music... But it does change Beethoven.

Beethoven was a republican, highly interested in liberty and equality, ideas that continue through many of his works. While the music isn't liberated by the Enlightenment, Beethoven liberated music by placing the composer as the creator and artist rather than the artisan. The coming Romantic Era places the artist at the center with characters like Liszt and Wagner changing public perceptions of the composer.

The songs you mentioned are the opposite of the Classical music of the era. They aren't written for the nobility that is paying for a finished product, the people are making songs for themselves. This is the big difference, as Classical Music is generally for the nobility (at least standard concert pieces) while La Marseilles is a war song made by the common person. The two worlds are vastly different.