Wow a question I can actually answer to the fullest extent of my knowledge!

Please keep in mind this is from a western scope. I'm talking Europe only here.

The truth is that much of our knowledge of who exactly wrote something has been lost to time. Many texts or chants have survived (the famous 'dies irae' is believed to have been written around 1100, but even that isn't as old as youre asking for) but we just don't know who wrote them.

Additionally, most music at the time was not written down. Notation wasn't really taught outside of monasteries. So if you weren't a monk and you were a great composer, we will probably never know. Secular music wouldn't have a hold on society until much, much later.

Pre-1000 AD you have a huge culture shift across western Europe as well. Post-Charlemagne, Christianity becomes the standard and sacred music with it. If there were composers that the Roman empire or the Germanic tribes considered notable, their music would've been forgotten or deliberately erased by the church.

One last thing: because the church had such control over western European society, the music that DOES get remembered is all chants. And for a long time it was all monophonic, or in modern terms, a single melodic line with zero accompaniment, instruments, harmonies, backing tracks, etc. This is like a whole other topic but it was a huge point of contention in the church. Unless you really, really know chants, they are all going to sound the same. To write a new one isn't... that hard (that will probably make some people angry lol) but to write a one that is memorable enough to stick around for literally a thousand years is a difficult task.