This sounds like a job for a centrifuge so I wonder whether the sort of people who home engineered their way to dishwasher sous vide have come up with washing machine stock straining.
I've read some weird hacks but the traditional non-centrifuge way to clarify the stock is to use some protein like ground meat or egg white to "absorb" any discolorant, protein, etc.
First time I made beef stock, I had much more than I needed; so I decided to turn some of it into consommé. You stir beaten egg-white into the stock, and simmer for a few minutes. The egg-white magically gathers the particles in the stock, and floats to form a raft on the surface. You then lift off the raft, to reveal limpidly clear consommé.
My consommé was lovely; but no better than Baxter's tinned consommé (and no worse). I haven't tried again since.
I just use a fine mesh strainer. If you really want clarified stock, you can simmer it with some egg whites and/or shells, the proteins will agglomerate all the remaining sediment. Strain again and done.
one round through the sieve without the cheesecloth, discsrd large solids, and then one round with the cheese cloth, no squeezing. I don't typically add vegetables to my stock thought, as it seems to me to make more sense to me to add them later, so that the flavors don't evaporate off.