The cover makes me think that the users of this type of quiver were more interested in keeping their arrows safe and secure rather than having them quickly and immediately to hand. Maybe archers would have dropped this quiver to the waist when preparing to shoot but carried them on their back when on the move? The depictions on the pottery could be soldiers caught off-guard, or artistic error.
Given how
many consistent references we see in that video, I think it's safe to say the pictured quivers were meant to be carried and used on the back as depicted.
But a quick google image search of Greek pottery shows tons of waist quivers to - lots from eastern opponents, but also (I think) Greeks.
It
seems the waist quiver shows up in military contexts more often (I can't tell which side) with the back quiver more in hunting contexts - but that's a very small sample size, a very quick survey, and I know nothing about that part of the field, so I'm disinclined to attribute that observation to anything other than the random selection I saw.
Keep in mind that one difference between the two cultures is not just time but location. There may be a consideration for the weather in Northern Europe that forces archers to keep their quivers by their sides so the fletching can be covered by a cloak. In Greece, maybe it's not quite that wet.
I wondered at that question of location - it's also possible there's more underbrush in northwest Europe to snag your fletching on than in a Mediterranean climate, but anything I can think of sounds like a just-so story.
Lacking any reference, I'm leaning towards fashion and historical accident, but don't hold that with any strong conviction.
*whispers....Year of the Ranger....
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I smell it in the air...
Persona: Aerlinneth, Dúnedain of Amon Lendel c. TA 3010.