Oops! You may have chucked a good wheel away. The top one (called the guide pulley) has to have side-to-side play whereas the bottom one (the tension pulley) shouldn't have side-t0-side play.
The play is there to automatically fix gears that are just slightly out of tune. It therefore runs much quieter than a fixed-spindle pulley.
Good quality pulleys used to have brass/ceramic bushings that could be removed, cleaned out and oiled. Nowadays some of them have small cartridge bearings in there which are mostly rubbish. That is no place for a tiny ball bearing, a bushing performs much better.
This was years back, and if I remember correctly the jockey wheels that had the problem had a number of loose bearings in a groove that I was constantly having to disassemble to repack with grease. The ones I replaced them with had bushings and solved the problem.
Good point about the top wheel being designed to have some play in it though.