Hi,
I think the sideways play you see in the large gear, is it just moving on the shaft splines.
Pull in and out on the lower shaft at the wheel hub, if it is snug, with no end play, the bearings are not loose.
I have seen them that the large gear has the sideways play.
Below are pictures at TM Tractor Parts of the large gear, showing the splines.
http://www.tmtractor.com/tm-tractor/gfi/drgear2_001.htm
Below is a page from the Cub and LoBoy service manual showing the final drive lower gear, the assembly info doesn't mention anything about checking the side to side play of the gear.
https://farmallcub.club/rudi/farmallcub.com/rudi_cub/www.cleancomputes.com/Cub/Blue Ribbon Service Manuals/GSS-1411 Service Manual for Cubs and Lo-boy Tractors/Section 7 - Final Drive And Brakes/Page-07-07.jpg
If I'm wrong, someone will probably say.