I think you're confusing "influential at the time" (quoted from your question) with well-known today. Popular music, meaning music that is played and heard in homes and taverns and pubs and bars, enjoyed by the majority of people, affordable and accessible, and usually featuring vocals and topical lyrics, was very influential, at least as far back as the 1700s.
This site lists a fairly large number of 18th century songs and it's only American songs: http://www.americanrevolution.org/songs.php
This article talks about the influence of popular music in 18th century Britain:
...pubs and taverns tapped into the mania for music, devoting upstairs salons to public performance.
there was still music to be had on every street in Britain. The milkmaid, lavender-seller and the knife-grinder heralded their arrival with a well-known tune; local waits, or pipers, played for every christening, festival and funeral in town. The streets of Covent Garden in London teemed with ballad-sellers and dingy print shops, producing thousands of new songsheets every day. The sellers, mostly impoverished women, would cry out their wares in song, and everyone from a lord to a country lass knew the tunes by heart. Freshly minted lyrics would tell the latest news of royal affairs, domestic scandals, ships lost at sea. A song cost a penny and once the words were out of date, you could use the rough sheets as draught excluders or toilet roll.
Popular songs from the 18th century and earlier that can still be heard today include:
- "Yankee Doodle"
- "Greensleeves" (first published no later than 1585!)
- "Ring Around the Rosie" is thought to be from 1790
- "Danny Boy" is sung to a tune that seems to have first been published in 1792 as "Londonderry Air"
Popular music was very important throughout the Renaissance, Baroque, Classical, and Romantic periods, and there are two reasons I can think of why it is not necessarily studied or heard more often today:
- It just didn't survive. The music that we study today was the music of the rich, the church, and of royalty. It was expensive, it was written on the finest manuscript paper or parchment, it was published extensively, and performed in the most lavish places for the richest and most powerful people. Or it was liturgical. Either way, its preservation was desired by those who had the power and money to preserve it. Popular music was more often of the moment, and when a particular tune became old and no longer exciting or relevant, it was no longer published and those who had purchased it did not go to great lengths to preserve their copies.
- It wasn't necessarily as musically complex. By the time you get to the Baroque period, the music composed by the great composers was complex enough that today we could study merely Bach's fugues for years and still uncover interesting aspects of them. A lot of popular music of the time recycled the same simple melodies. Since there were no recordings made, if you wanted to hear a song, you either had to pay someone to play it or play it yourself. That meant that the popular music was normally much more straightforward and simple. In music theory classes, there's just not as much to teach about such music.
We can't really say what the future will hold, because prior to 20th century recording technology, it was much harder to preserve popular music, and it was more difficult for a popular music writer to become fabulously wealthy by selling music to common people (as opposed to the rich and powerful). The music of The Beatles (for just one example) will almost certainly be much more available for study and enjoyment in 2163 than the popular music of 1818 is today.