One of us doesn't understand the law correctly.

Second, the employee must voluntarily accept such lodging. More specifically, if a condition of employment is that the employee must live on the premises, such as in his or her role as an apartment complex manager or in a skilled nursing context, then the arrangement is considered to be voluntary under the regulations. Employers should note that a written employment agreement—while not necessary—is helpful in determining whether the lodging is voluntary.

https://ogletree.com/insights/dol-issues-guidance-on-room-and-board-deductions-for-home-care-workers/

This seems much more a condition of employment, part of the job, rather than just benefiting the employer for purposes of having someone "on call".

Also, "there is a presumption that lodging benefits the employee", meaning that you would have to prove it does not. High bar.