"I wonder if Mom paid capital gains tax on the arm's-length selling price."

No sale. Mom gave it to the kids.

But what a mess has been created.

Properly drafted Trust would have been better - Grantor Trust for income tax purposes and step up on death.

And wouldn't this gift to the kids who put it in an LLC be subject to a Medicaid clawback or whatever it's called? Or are they betting Mom outlives the clawback period?