Executive privilege

Neither executive privilege nor Congressional presidential oversight are in the Constitution, so their authority is established through precedent. Both are the subject of court cases being heard today - Kupperman's duty to obey a subpoena from Congress vs Trump's orders to not appear; and Don McGahn's long-running refusal to testify before Congress regarding the Mueller investigation.

My understanding of executive privilege is that it applies to maintaining the confidentiality of conversations between the president and immediate advisors on issues pertaining to national security or the ability of the government to function properly.

While a case could be made that "taking down" a president is a matter that probably would have the effects listed above, if the confidentiality relates to wrongdoing then it stands to reason that any secret information needs to be revealed precisely to improve national security and proper functioning of the government.

Surely EP does not apply to covering up presidential crimes. But what will the courts say?

You never change things by fighting the existing reality.
To change something, build a new model that makes the old model obsolete.
R. Buckminster Fuller