I believe that by the 5~6th centuries Animal powered Artillery had reached it's limit of development.
The Medievalists did come up with the Trebucket, but that too rapidly reached state of the art.
New materials could make some minor improvements, but those were usually adopted OTL as soon as developed anyway.
The Bow also had reached it's limit, whether the English Longbow, or the Mongol Shortbow. Both took time to learn, and time to develop the necessary muscles to maintain a rate of fire.
Perhaps if some pre gunpowder genius comes up with the cammed Bow, whe can delay the Gun, as it's early use was considerd slow, clumsy and dangerous.
But once Gunpowder arrived on the battlefield, its use in Sieges, can't be stopped. From there it's use is going to expand.