You couldn't be more wrong. The President is the head of the executive branch. If he legally orders all DEI programs to end, they end. If an agency, or persons at an agency try to hide and continue such programs, they are in fact in violation of federal labor law and civil service standards. They should be reported and if found to be doing that, disciplined for ignoring management's orders.
That isn't any sort of "spying." It is the duty of an employee of the federal government to report those that are openly defying their leadership and management. That goes for anything, not just DEI.