Auto Home not close to bed.

Hi, I know this question has been asked in related ways before, but I just want to get confirmation of the principle. I just started using my son's Ender 3 Neo for my first project (he went off to college). I am intending to level the bed, and I start with Auto Home. The head moves to the center of the bed, moves up and down a few mm, and then stays about 8-10 mm off the bed and waits. I have done the Moving Axis > Move Z adjustment to get it one paper away from the bed (including also turning the leveling knobs), I have then done the Z offset (-02.500) to get it one paper away from the bed. But then I hit Auto Home again, it ends up in the same location quite a bit from the bed (and the screen says Z = 5.52). Is it supposed to do that? Or should Auto Home take it to within a paper of the bed?

(PS, I'm not sure if the Ender Neo counts as Ender3Pro, happy to delete this post if it violates rules).