You know, I like that we are a people that would put so many resources into an attempt to find these kids, as unlikely as it was to be successful. I think it says something good. And every now and then we'll have it end in a "miracle": Whether it's kids lost at sea, miners in a collapsed coal mine, sailors trapped in a stricken submarine, etc, every now and then we'll succeed.
Anyhow your idea is interesting. Consider the opposite extreme to what normally happens: Suppose NO govt money could go to efforts like this, and it had to be up to the public to send in donations to search for these kids. I believe millions would have poured in -- probably at least a dollar for every tenth person in this country, so tens of millions. Still there could be an issue on day 1/day 2. But your combo idea would allow for govt resources to be there on day one and day two, then when private funding had built up sufficiently it could fund several more days and more extensive searches. It won't happen I don't think, but it's interesting.