Not sure how it works for CIA and senior executives but for the military you lose your Top Secret Access when you leave a billet that requires it. For code-word accesses you get “read-out” in a meeting in which you are instructed on the rules/laws about any type of disclosure. If you have a TS Clearance it expires 5 years after you last investigation completion unless your organization updates your background investigation. Also, to get access you have to be certified as having a “need-to-know” by a certifying authority.
One would hope... that John Brennan lost his access when he left the CIA already. Pulling his clearance for cause will make it difficult for him to get it up dated.
The president is in control of all clearances at every agency and level of the government, period. People here used to ask “how in the world did Barack Obama get a clearance?” He didn’t need one. He was the authority in charge of such matters, just as Trump is now.