New member, transmission question

I have a 10323 self propelled mower that i am referbing.

I took all the stale grease out of the transmission and re assembled it with new grease. after i put it together, i noticed that there is quite a bit of slop in the pulley. I pulled things back apart, seems like the gears will only go in one way, so..... That leaves the bearing that is on the pinion gear.

So, are these things supposed to have that much slop, or is it time to punch it out and replace it. While apart, i did inspect it and noticed it did feel smooth when rotated.