Q. Since his ROOM was paid with these taxable funds, can I remove an equal amount from a 529 tax-free?
A. Yes. I know that may disagree with @Carl answer.
The school applying the taxable scholarship to room and board (R&B) does not lock you into saying it was used for R&B, at tax filing time, unless it was a condition of the scholarship*. You may withdraw that much from your 529 plan. You don't actually have to pay R&B with those specific funds. You only need to have that much R&B expenses in same tax year as the 529 distribution.
I agree with your analysis, once the scholarship is taxable, it could be used for anything - beer, football tickets, car expenses. You are I interpreting this correctly. One way to think of it: you only borrowed money from your son's taxable scholarship, to pay R&B, until you reimbursed yourself with 529 money.
*Although some (a few) scholarships have condition that the grant be used for tuition and fees, a R&B condition is even rarer.
