I have 4 different low shank Kenmores that are center homing & don't have moveable needle positions....so no LCR lever. Of course, if it IS left-homing, the moveable needle positions solve that!
I have an early 385 Kenmore that has a left-homing zigzag....in other words, the left side of the zigzag stitch is always at the far left of the foot....& as you decrease your zigzag width, only the right side of the stitch moves. But if you're straight stitching, the needle is centered.
A lot of the super-high shank model Kenmores are left-homing....maybe all of them are. None of my low shank Kenmores are. But, that's not to say that none of the low shank ones are...just none of mine!