I have read somewhere today (probably on NRO) that the debt limit is set by statute and that an Executive Order cannot override statuatory law.
The EO option doesn’t exist.
The other objection is that the 14th Amendment deals with “public debt, authorized by law”. If the debt limit is raised by EO, then it is not authorized by law. Would you buy debt instruments which are not authorized by law and therefore may not be re-paid?