If a creative work or research endeavor costs $10,000 to produce, and public grants provide $1, does that dollar buy the public unrestricted rights to the work?

It sounds like you think it should, which is a fine position to take, but odds are that the contributors of the other $9,999 have a different opinion and may make different choices about contributing were that the case.

So — assuming that the public can’t justify the entire production cost of all grant-worthy work — what ends up happening is that you have a negotiation of rights and contracts and lawyers and all that banal, complicating stuff.