As you say, in theory.
In practice there is no such thing as full independence when it comes to senior appointments, whether they be in the public or private sectors. In the tangled web of science, even if they do not seek only to advance their personal careers, the individuals concerned must play political games if they hope to achieve anything in their jobs.
In light of the EPSRC's decision to kill off the PPE scheme, maybe it's time to formulate some very specific questions for our political leaders and their executive agents in the research councils. Questions that cannot be answered with meaningless PR spin.