Can only guess but it still seems like a bearing. While hoisted spin the wheel - it should be near silent. It can be a failed bearing long before it show "freeplay"
Also take a good look at the tires. If feathered wear and just balanced some can make noise and can be worse when rotated to opposite side of vehicle noise wise if not suggested for the specific tire don't do that.
Shake in steering suggests tire troubles again or perhaps a bent rim. Did they rotate tires/wheels when they were balanced? The problem may have moved if a tire or wheel and should have been noted when balanced,
T
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Tom Greenleaf - MetroWest, Boston