I've got the COFI as well and the answer is No, you will not get clear wort from recirculating. Before I got the BB I was mashing in a 10G round cooler with a stainless supply line braid and would vorlauf a couple times while batch sparging. That would result in a very clear wort into the BK.
With the COFI I was expecting similar or better results since it was constantly rinsing the grains and I figured it would also "set" the grain bed. However the wort is very cloudy and even has some grain particulate in there. I was pretty bummed since I had added a BrewHardware flow view sightglass and was hoping to see crystal clear wort being pumped through.
However, after reading up on the COFI and BIAB in general, I found out that super clear wort is not to be expected and that cloudy wort doesn't really matter. My beers made with the BB have still cleared up nicely, both with and without using gelatin, and they have been just as good or better than on my old system. I still add a whirfloc tablet with about 10-15 minutes left in the boil and I use hop sacks (mainly because I have a plate chiller now), but I don't go out of my way to keep break material, grain particulate and other gunk out of the fermenter.
As for cold crashing in the keg. The advantage of cold crashing the beer before transferring to the keg is that a lot of stuff drops out and sticks to the bottom of the fermenter so it never makes it to the keg in the first place. Additional time lagering in the keg will get some more stuff to settle, but in general I find it more effective to chill the fermenter before transferring to the keg if you have the means. May still be a little cloudy, but clears up a lot faster.