They need a regular barracks assignment as well since they do some non-archery training as well.
I question that. I had two squads and two separate "barracks". One of them had weapon racks and armor stands, the other had archery targets (and
no weapon racks, armor stands, or even coffers or cabinets). I made the archery targets, designated
each one as a room assigned for the training of my marksdwarf squad, and set an appropriate firing direction; and I made a lot of wooden bolts, to be stored in the adjacent ammunition stockpile. The other barracks was
only used for the training of my axedwarves. My marksdwarves were in fact seen target-shooting.
Since you mention assigning ammo, you presumably have plenty of it. I'm not sure what to make of the abandoned quivers, but I
think that when shooting at an archery range, dwarves
do not shoot from their quivers, but drag a pile of bolts with them. (Their quivers are for combat-ready bolts that they will take with them.) This may get more complicated if combat and training bolts overlap, but I'm pretty sure I've seen them hauling bolts.
Check your archery range. Make sure that all the targets are made into rooms (not just one) and assigned to the marksdwarves, and that the firing directions leave enough room. While I'm not 100% definite on the carried-vs-quiver'd bolts, I'm very certain that I didn't need to assign my marksdwarves a standard barracks in addition to the ranges. You say you assign "one per squad", which makes me think that they may want to train, but someone's already using the assigned range, so they can't. Are you sure, thus, that there are absolutely
none of them training at the range? (See UdV's comment about having one range
per squad member.)
And what kind of bolts are in the quivers, anyway? Are they training-suitable, or combat-suitable but not allowed for training, or something else entirely?
I'm sorry I can't be more definite, but I struggle to get a military equipped before the fort gets overrun, so I rarely have time to play with the military in detail.