I wrote a reply on this a few months ago that may help to address some of your questions. Take a look and I hope you'll come back if you have follow up questions or disagreements!

Here's the key part of the top-level reply:

No, they were just never very popular. Hexameter wasn't a very prestigious genre of poetry until the late 500s BCE: if you think of poets who were invited to take up residence at a tyrant's court, they're always choral, melic, and elegiac poets, never epic poets.

Homer's name was known prior to the late 500s, but the main bit of testimony we have prior to that date links him to the Thebaid, not to the surviving epics. The popularity of the surviving epics doesn't seem to have taken off until the late 500s, perhaps as late as the 520s. They got lucky. The other epics ... just never got lucky.

A metric for the change in the Iliad's popularity in the 500s is the prevalence of scenes in vase paintings. Vase painters weren't menial illustrators, and they had their own stories, but their choice of scenes is still telling: prior to the mid-500s, scenes from all over the Trojan War appear in vase paintings. But from around the 540s onwards, scenes related to episodes in the Iliad suddenly shoot to prominence and account for over half of all Trojan War scenes. That strongly suggests that prior to that point, either (a) epic poetry wasn't an especially dominant medium for these stories, or (b) all the epics were equally popular.

And then the Iliad suddenly swamped popular consciousness. It got lucky. Maybe it was a lot better than the other epics, and that helped: that's what Aristotle's claims. But Aristotle certainly isn't infallible, and we'll never know for sure. Anyway, if that sudden rise to fame hadn't happened, the Iliad and Odyssey would certainly have gone the same way as the other epics.

We can count on one hand the number of extant ancient writers who claim to have read any of the lost epics: Herodotus and Aristotle in the 5th and 4th centuries BCE, and Pausanias in the 2nd century CE -- and some scholars don't even believe Pausanias. And Pausanias only claims to have read two of the lost epics, the Cypria and the Little Iliad.

Available evidence tends to suggest that, apart from the exceptional role of the Iliad and Odyssey from the 520s onwards, people didn't get their knowledge of the Trojan War from epic poems. By the Hellenistic period, they were getting their info from from prose accounts. You're probably aware that we have prose summaries of the lost Trojan epics: that seems to be what this material looked like.

The prose summaries, and the account in the Library of pseudo-Apollodorus (one of our main sources for Greek myth), are based on the lost epics: well, the evidence points to people reading these prose versions and not reading the epics. When poets like Vergil and Ovid alluded to cyclic episodes, there's no particular likelihood that they knew the poems directly: they were very probably working from prose summaries, like the ones that we still have today.