I use shooting circle all the time in my Byz game, despite the relative lack of accuracy and fire rate :)

My main reason? I usually deploy all cavalry armies for field battles, so I am not using Ha to support/protect an infantry line.

So, say I'm facing a typical Hre/venice/milan stack of 6 pavise xbow, 6 knights and balance of general/cav/siege etc. My stack will be 8 or so vards(and a few high xp HA leftovers), general, and 8 or so units of kataphractoi/latinkon heavy cavalry.

I will first send all my HA to encircle the enemy on shooting circle mode. I stagger deployment to give each unit plenty of room to work in, and leave autofire on once targeted.. This means they take little to no damage from enemy missles, while inflicting reasonable damage - a gold vard is fairly accurate even in shooting circle mode, whereas green troops are fairly rubbish.

I always target enemy missles first, and will break troops off from the shooting circles to engage any enemy HA/cavalry that leave the army with a loose skirmish formation. Whilst my HA is doing this, I move all heavy cav to a hilltop and keep away from enemy missle/siege.

As each Vard/Ha unit runs out of arrows, I move them out of missle range, most will be at or near full strength. If my whirl of shooting circles has spread the enemy all over the battlefield, it's an easy job for cavalry to mop up individual units with minimal losses. I then use depleted HA to chase down routers and secondary/flank charge my cavalry targets.
If the enemy remains in a 'bulk' then they will generally take 30-40% loss before my cavalry engage in a massed howling charge supported by all my HA 'medium Cav' and usual dreadlord 'lead from the front' general.

Overall, the circle allows all my troops to fire all of their arrows badly, then act as near full strength cavalry. In non circle mode, you can easily lose half a unit if they sit for a few seconds too long, even if firing more accurately etc. not good when they have plenty of ammo left..