The Greeks (we'll just talk about them since the Romans more-or-less adopted the Greek literary and philosophical canon) did not look to Egyptians or Sumerians in a way that is analogous to the European tradition of Classical Studies. The Greeks really had very little awareness or even opportunity of transmission for this. They did have significant connections, however, but they looked elsewhere for literary inspiration.

Greek literature is deeply rooted in their oral storytelling traditions, which date back to the Bronze Age in some cases. Those traditions were monumentalized into "great works," most notably Homer's Iliad and Odyssey, which became foundational. These works, however, only record one tradition of the stories that Greek parents would tell their little greeklings before bed (this variation manifests itself in all forms of Greek art and literature-- see Iphigenia and Greek pottery for two good examples).

So, for the Greeks "the classics" did not come from a different "older" culture, but rather out of their own oral traditions. (It is of course way, WAY more complicated than that, and I encourage you to explore the Bronze Age Mediterranean and subsequent Dark Age (I also encourage you to avoid sources that draw a 1:1 comparison between Bronze Age Greece and Homeric Heroes)).

The Romans, as I said above, mostly adopted the works of the Greeks (though they both developed their own admirable forms of literature and thought as well as expanding upon the Greek literary canon). Aristocrats would send their sons to Athens or Rhodes to learn from the great orators and scholars of the day or buy Greek slaves to teach in their homes. I suppose adoption at the point of the sword should be better considered appropriation, but no matter... A great example of this is Vergil's Aeneid, an ostensibly political epic poem commissioned by Augustus to help legitimize his new regime by connecting the Roman origin story to Homer's heroes and myths.

The Romans, nonetheless, differ from Medieval and later Europeans in that Greece was a center of learning and art contemporary to the flourishing of Roman culture, as opposed to "rediscovery" on a monastery shelf. A better analogy would be something like sending your kid to Harvard as opposed to PoDunk U, because it is a better education.

Finally, I would be remiss to mention that "the classics" is a pretty loaded term, that is not derived from any sort of intrinsic value of the works of Greece and Rome (though I do love them personally), but rather from the idea that knowledge of them became a marker of a certain class (from the Latin classis).