I think you are on to something here.  This benefit is more than incidental.  It’s not like a free lunch at the noon board meeting or free attendance at a ticketed event where they are performing some official function for the organization.  The IRS is likely to consider it compensation.
If the IRS looks and the directors have not treated the benefit as compensation in the year they received it, the IRS could call it an automatic excess benefit.  (See Ready Reference Page: “IRS Issues Tips to Agents on Collecting ‘Automatic’ Excess Benefits Taxes.”)  Then they would not only have to pay the benefit back to the organization, but would also have to pay the 25% excess benefit tax and the late reporting fee.  Any board member knowingly approving the benefit might also have to pay a tax, and a person signing the Form 990 tax information return without reporting it might also have a problem.
Since the compensation would not seem to be unreasonable—and therefore not itself an excess benefit—it may be a lot wiser to treat it as compensation and have the directors pay the income tax.  They would still end up paying a lot less out of pocket for their kids’ participation than the general public.