Check out this weapons video to see how to prime .223 brass with the Lee Turret Cartridge Press.
You can prime brass immediately after resizing and depriming, which could be a real time saver, but for rifle brass you'd have case lube on sized brass that you wouldn't want to tumble to clean because the media would clog the flash hole. Even if you resize and deprime, then tumble the cases to clean them, then trim and chamfer them, and then run them through the Classic Cast press again to prime them with the Lee Safety Prime, it's still a fast way to prime brass. You also get a good feel for the priming operation, so you can feel loose primer pockets, as well as primer pockets that still have a crimp that needs to be removed. It's a good way to prime brass, and you'll like it. You may stop trying to maintain the fussy priming system on the Lee LoadMaster and prime all of your rifle brass prior to the progressive loading on the LoadMaster.