Rear wheel causing road bike to vibrate
Yes, a hub that's loose and torn up inside can cause what you're experiencing, and given the evidence you present sounds pretty plausible.
Your experiences with the shops suggest that one or more of your bearing races (cups and/or cones, typically starting with a cone) are damaged. The one that recommended a wheel was cutting to the chase. The one that adjusted it did so because adjusting out the looseness of a damaged cup and cone hub can limp it along, but the nature of cup and cone hubs is that a good adjustment becomes impossible once there's any damage, and the damage perpetuates. So they tightened it, it got more torn up inside, and now it's worse - hence the virtue of cutting to the chase.
If it were just a problem with the cones, they can sometimes be replaced on an OEM wheel such as this. Having a shop do that plus the overhaul labor is often iffy value because the cups, which aren't replaceable, are likely to have picked up some damage at this point too. It's perfectly likely they're heavily damaged, in which case going straight to a new wheel is the only reasonable option.