Seems to be working now
Well, I tried it again today and everything is working smoothly. I was able to set up the group, add my wife, and create accounts for 3 of my kids. I did get a new phone number today but I don't see why that would trigger the reset when it's still the same iCloud account. My guess is that it just took a while for it to roll over. If the latter is the case, I'm not sure why Apple Support didn't share that or maybe they just didn't know. I'd recommend others having the problem to give it a try again and see if it works.
As to mantan's comment about usefulness - I thought the same thing in terms of buying apps. Why not just use individual Apple ID's for each person and a separate purchasing ID? Well, with this sharing, you can indeed do that and more. When you set it up, it asks which iTunes account you use for purchases. In my case, that's a different Apple ID from my personal Apple ID. In other words, when you set yourself up as the owner/administrator of the family group, it attaches your Apple ID and, if you want/need to, a separate Apple ID just for purchases.
The advantage then becomes as you have kids who have to buy their own apps and music - they can share that music as long as their a member of your family group but when they go off and start their own family group, they can take all of their purchases with them. Where it perhaps falls down is that if there are apps/music/movies that you'd like to keep when the kids leave, you'd have to re-purchase them if they were purchased with your kids' Apple ID. All paid for by the same credit card but connected technically to the individual account from which the purchase was initiated. At least that's how I understand it.
In addition, one HUGE benefit I see is the shared "Find my iPhone." This way, if I lose my iPhone or my wife loses hers or any one of the kids lose their devices, we can use any other device to locate it.