If the airline commits contractually to supervising children as unaccompanied minors (in return for a fee, of course), then they become the airline's responsibility during the term of the contract of carriage.

Nothing obliges airlines from accepting UM's at all. They could simply refuse to fly them outright by further modifying their COC. As long as they do accept them, subject to their T&C, then UM's are most definitely the responsibility of the airline.