Government agencies in the executive branch don't have independent authority. They work for the president, and an EO isn't much different than the email you get from your boss directing you to do work a certain way.

An email from my boss telling me to implement something using rust transpiled to wasm certainly impairs my authority to determine the best approach.

My argument is not the EO has the legality to make a claim; it's that the top half of the EO is at odds with the disclaimer at the end. If you mandate somebody to do something then you're impairing their authority to have chosen not to do something.

Like by definition the EO impairs agencies that were using their authority to issue paper checks to continue doing so. It may be advantageous to stop issuing checks but to claim mandating that they don't doesn't impair their authority is just false.

You're conflating authority, the authorization to do something, with autonomy - the ability to use that authority at your own personal discretion. The law grants a regulatory agency the authority to do something that they would not otherwise be able to do, like nationally ban widgets. But the law does not also inherently grant them to the autonomy to do so entirely at their own discretion. For agencies under the Executive branch, the President is free to direct them to utilize their authority at his discretion.