I know you mentioned cold crashing, I do cold crash to about 35F over two days before kegging and usually fine with gelatin about half way through the cold crash.
I usually clean a few kegs at start of filling operation. When a keg kicks I pull it from keezer but keep it pressurized. Then on kegging day I clean all dirty kegs. Disassemble posts and poppets, rinse well, run on Mark II keg washer with hot PBW for 10 min, rinse well, reassemble. Some of the steps can be done in parallel but if I have 3-4 kegs that need to be cleaned this process will take up to an hour.
Once I have two clean kegs first is filled with starsan. Usually do make a new batch which is a few minutes. I then use combination of gravity and CO2 at about 5 psi to push the starsan from keg to keg. This takes about 10 min per keg. I'm now about 90 minutes into packaging day and have 3 purged kegs and probably a fourth clean keg full of star-san.
While those are going I will also get my beer transfer line ready and sanitize fittings between fermentor and transfer line. Then hook up pressure to the fermentor, give a bit of pressure and start transfering. If I am very lucky this will take about 20 minutes per keg. If I get a clogged poppet it will take much longer as I will have to stop the transfer, disassemble the poppet, sanitize, reassemble and start transferring again. All while cursing the fact that my closed transfer has been compromised and knowing full well that if it clogged once it is probably going to clog again. Most beers I make I get a clogged poppet. These are typically ales with about 6oz dry hop in the fermentor. I do try to turn my diptube to be pulling beer from above the trub but doesnt take much of that hop material to clog the poppet.
So anyway on a good day the filling of the kegs is a 60 minute project and I'm at 2.5 hours. Then I harvest or dump the yeast and clean the fermentor. By the time everything is clean, packed and put away, full kegs are loaded into keezer and hooked up to pressure I am over 3 hours and if I did a good job with cleanup probably over that.