how do I remove partially-disintegrated sediment/tiny-bones from stock?
Straining might work, but you may need to use a process known as "decanting":


Let the stock sit until any sediment falls to the bottom.
Remove the good liquid, avoiding the sediment at the bottom.


You can do this a few ways : 


Use something to scoop the good liquid off the top
Use a hose to siphon off the good liquid until just before you get to the sediment
Carefully pour off the good liquid, stopping when you notice the sediment rising to the top


This is more difficult with stock, as when it cools, the gelatin will start to thicken making pouring a bit tricker ... but it means one more potential method:


Let the stock set up, then remove it as a block from the container (you may need to set the whole thing in hot water to get it to release; sometimes a hot-water spray or wrapping it in damp hot towels is enough).  Slice off and remove the sediment layer.  Put what's left back in the container.