The President does not have the authority to rewrite a law.
True. But Executive Orders can tell the employees of the Executive Branch how to respond to certain situations. He can order a department to reconsider a rule in many cases. The "deep state" was (and to some degree, still is) comprised of Executive Branch staffers exceeding their authority by writing rules when Congress should have stepped in - but didn't, the dirty boogers. The president can make the deep state bureaucrats stop enforcing rules that didn't originate from Congress. Take a look at his recent order to protect the 2nd amendment that is causing the DOJ to reconsider a lot of rules promulgated illegally by ATF.
Consider the Dept. of the Interior and their Wetlands rules, and the ATF and their firearms rules, which were DEEPLY derived from rules created by Executive-Branch
fiat during the Obama and Biden administrations, and that Trump's first term was marred by insane levels of TDS, there are literally hundreds of Executive-branch rules out there that are being enforced as law - and in some cases result in criminal convictions - that shouldn't even be on the books. (See also the Chevron Deference Doctrine, now defunct thanks to the Roper-Bright case.)
Specific case in point: ATF's silencer and short-barreled rifle laws, the latter being applied to braced pistols. Specific case in point: Wetlands classification rules that have the result of an uncompensated taking of property by denying the owner certain uses over standing water that doesn't enter into a flowing waterway. Specific case in point: DWF requirement for fishing boats to have an inspector who will not help with the fishing enterprise and who must be paid by the boat's owner. Non-productive overhead.
That's three examples out of a much larger bunch illustrating Executive Branch rules created by
fiat that were not acts of Congress but that resulted in fines and criminal charges. If they weren't laws, they had results that remarkably resembled laws. So when you make your claim that the president cannot rewrite a law, be sure to distinguish between an actual LAW passed by Congress vs. an Executive Branch rule that, through its consequences, has the EFFECT of law.