In Scenario A, no, it only removes memories at the time of death. We see this with Yaana and Neyuni in the Arcadion questline. Their parents taught them not to wear regulators, but Yaana and Neyuni both remember their parents even after they started wearing regulators.

In Scenario B, also no. In the Arcadion questline, the President is said to have taken the regulators of "immortalized" fighters so that they die of psychonekrosis still remembered by all of their fans to keep up the illusion that they're simply living in luxury on the upper floors of Everkeep. Both parties must be wearing a regulator at the time of death for the memory erasure of the dead to occur.