The lube sticks are designed for lubri-sizers. The Lee sizer dies are designed to be used with a tumble lube, like their liquid alox. Any style cast bullet can be put through the sizer dies and tumble lubed (or pan lubed), but the bullets designed for tumble lubing do not do well in a lubri-sizer. For those bullets with a standard lube groove, it is not necessary to fill that groove if tumble lubing or pan lubing. In fact, with the LLA, it is detrimental to fill the groove and you will get heavy leading and smoke. With the tumble lubes, you only need a very thin coat over the bullet. The mix I mentioned above is all I use for handguns (including the .357mag) with velocities generally staying below 1400fps. I get virtually no leading if I have a properly sized bullet.
With the lube sticks, it is important to fill the lube groove because you don't have lube on the entire bullet. The driving bands do not have lube on them when shot, so you need the extra lube in the groove to prevent leading.