This doesn't eliminate the penny, just pausing production of it. If he had unilaterally declared the penny no longer valuable then that would be unconstitional and require an act of congress
The distinction you make is irrelevant since both are under Congress' responsibility. The president has no constitutional authority to determine which currency is minted and how much of it is minted.
This law does not mandate continuous minting. So I think you are wrong that the Secretary of Treasury does not have the ability to say how much of a coin is produced day to day.
Legal scholars, such as Laurence H. Tribe, have noted that U.S. law grants the Treasury Secretary the authority to determine the quantities of coinage necessary to meet the needs of the United States. Therefore, if the Secretary decides that the required amount for a particular coin is zero, they are within their legal rights to stop its production.
