I'm going to narrow the focus to classical music, because it seems like that's what you're going for here. Many songs from different genres and traditions have had more easily remembered names.

And, for that matter, the vast majority of classical "songs" (musical pieces with a sung text) also have names, often based on the first line that the singer sings. So, for example, this aria from Mozart's opera Don Giovanni is commonly known as "Madamina, il catalogo è questo", or often simply "Madamina." Schubert's many songs pretty much all have easily-remembered names, as do the arias from Bach's cantatas or Handel's oratorios.

What we're really talking about, then, is instrumental music. And that gets to an important split in the classification of classical music. The boundaries are by no means clear-cut, but this is a topic that has been highly contentious in the past, with different composers arguing over which approach to the music is better.

One is so-called "program music." This is music that follows some kind of storyline, or takes after a specific inspiration. A great example is Berlioz's Symphonie Fantastique, which follows the opium-induced hallucinations of a lovesick artist, including a final descent into Hell after murdering his beloved (As a side note, Berlioz once meticulously planned out and almost executed a murder-suicide, around the time he worked on this).

The other is "absolute music," which stood on its own. That's not to say that the music isn't inspired by things from the composer's life, historic events, or literature, just that the music as written is not bound by that inspiration and therefore could and should be judged on its own.

Prior to the 19th century, most instrumental classical music had been more or less absolute music. Again, there are definite influences and inspirations, but the music was presented as it was. Hence "Symphony No. 1 in E Flat Major," which describes the form quite well but doesn't say much about the content.

In the 19th century, composers began to push back against what they perceived as the rigidity of the prevalent forms of classical music (symphony, concerto, sonata, etc.). A common contention was that "new wine needs new bottles," an argument for new forms to go along with the new ideas of the day. So, in the 19th century, we see many, many more pieces with unique names and forms, rather than simply "Concerto No. 3 in C Minor."

Interestingly, many publishers and critics did understand that easily-remembered titles are, well, easy to remember. And so, even though they were not named by the composers themselves, many classical pieces have unofficial names. Take, for example, Beethoven's "Moonlight Sonata," so named by a critic years after Beethoven's death. Or, Mozart's "Jupiter Symphony," whose nickname became popular decades after Mozart wrote it.

I am not a historian, but I do have a non-classical related degree (which included a fair bit of musicology, although I tended towards the practical side), two A-levels and a preparatory diploma in music, along with the usual grade 8 in an instrument or two, and theory etc, so consider myself at least somewhat qualified to attempt to answer this, but feel free to delete as I do not have any citations.

You are asking the wrong question.

The important thing to understand is that classical and popular music are two different things, with different traditions.

We tend to think of classical music as the music people listened to in the old days, but this is untrue. It was* the music of the aristocracy (and church, but that can be included in aristocracy, so meh). This had/has a formal, intellectualized tradition and structure.

On the other had, popular music (popular = of the population, rather than well liked) comes from an oral tradition. It is not necessarily less formal - there are generally far more stringent rules, but the rules are generally not written down because, for starters, often the people playing and singing it were illiterate, and their songs and performances were meant for a largely illiterate audience seeking entertainment rather than satisfying some kind of intellectual need (whatever means, or need may have been). The audience were there to be entertained and "a free-form jazz exploration in front of a festival crowd" was not the order of the day.

So, songs had "easy names" as they were not generally intended as an expression of something intellectual (for lack of a better word), but as entertainment.

All this started to change with the rise of the consumer society, a more educated/literate population, the invention of radio and recording devices, and later when it started to be taken more seriously at an academic level. So we see more of a merging of the two traditions with the result that popular music is taken more seriously, classical music is more accessible, and popular music has become more than just something to stomp your feet to while downing flagons of ale and/or banging whores, or waiting to anyway.

Radio and other media (and copyright law) and rising wealth also made the name of a piece of music an important marketing tool. "Smack My Bitch Up" is inherently more marketable than whatever a concise technical one line description of it would be.

So, to answer your question. Popular music has always had "easy names", but this became more important and prominent in the early 20th century due to technology and rising wealth. Classical music has never been popular (in the of the people sense) and serves a different purpose.

*And arguably still is - in the UK the proportion of classical musicians coming from the highest socioeconomic backgrounds is/was higher than any other profession (unfortunately I cannot find the article, but the premise was "are politicians overwhelmingly from higher socioeconomic backgrounds" - not as much as you would expect - and classical musician was one of the comparisons - scarily high, like 85-90% or more IIRC).