I don't think this is quite right.
The CFF was effectively started by the 45 Labour Government, on the recomendation of some bods who thought children's cinema entertainment was lowbrow and american. They needed better. But it wasn't in any way cnsciously leftie - all the kids had to speak in proper RP accents, most of the criminials they fought against were thick working-class types. Not exactly revolutionary stuff.
The Foundation itself was funded through the 'Eady Levy' which took 5% of ticket prices for redistribution to producers to makle more British movies. CFF got a cut of that money (it also went to a whole bunch of great foreign film makers to make stuff here - Truffauts Fahrenheit 451, Polanski's early work, Kubrick, etc etc). Many European countries had similar systems.
Thatchers government decided that the levy wasn't really doing what it was set up for in the early eighties (and - hate to admit it tho I do - they weren't entirely wrong) and abolished it, along with the 25% tax breaks available for film-makers. With the end of the levy, the CFF had no funding, and the fiulms they made weren't good enough to make a profit, not even with all those promo clips on Screen Test.