I'm not sure of the exact history, but it may have been copied from how affordable housing projects are subsidized at the Federal and state levels: https://sgp.fas.org/crs/misc/RS22389.pdf

Affordable housing tax credits are contingent on a tremendous amount of regulations and stipulations, including who ends up holding equity in the property (it's not easy to use these as passthrough vehicles to get personally rich), so the private lenders tend to be repeat players, often banks, who understand how to navigate the process. Among other things, I guess it's a way to outsource oversight, so that the government doesn't need to maintain a huge bureaucracy to police each and every development project. OTOH, the financing complexity comes at a cost; a significant fraction of the value of the tax credits pays for lawyers and accountants, rather than to actual construction.

I don't know if filmmaking subsidies are more lax or easier to game.

Also, I think using tax credit schemes, instead of direct payments, might be a way to obfuscate the cost of these programs from a legislative and political perspective; not unlike the Earned Income tax credit. Nominally speaking, tax credits reduce government revenues rather contribute to expenditures; the latter draws far more attention.