This is a logical fallacy known as ignoratio elenchi, usually translated “ignorance of the refutation”. Basically, even if everything in an argument is true, it still does not refute the counterargument. You persist in ignoring the statement in the code “according to Presidential instructions or orders, or in accordance with recognized customs or practices not inconsistent with law” in the code. Perhaps you are confused by the meaning of “or”.
or
conjunction
- (used to connect words, phrases, or clauses representing alternatives)
Therefore, “according to Presidential instructions or orders” and “in accordance with recognized customs or practices not inconsistent with law” are alternatives, indicating that sometimes it is presidential authority that orders the flag flown at half-staff and sometimes it is not. I don’t see why this is so difficult.
Second, the Flag Code only provides guidelines. It is not enforceable law:
Again, the provisions of the Flag Code on flying the flag at half-staff are, like all the Code’s provisions, a guide only. They do not apply, as a matter of law, to the display of the flag at half-staff by private individuals and organizations. No federal restrictions or court decisions are known that limit such an individual’s lowering his own flag or that make such display alone a form of desecration.
The United States Flag: Federal Law Relating to Display and Associated Questions
Congressional Research Service
https://fas.org/sgp/crs/misc/RL30243.pdf
Third, the VA takes its wording from Presidential Proclamation 3044 of Mar. 1, 1954:
5. The heads of the several departments and agencies of the Government may direct that the flag of the United States be flown at half-staff on buildings, grounds, or naval vessels under their jurisdiction on occasions other than those specified herein which they consider proper, and that suitable military honors be rendered as appropriate.
https://www.archives.gov/federal-register/codification/proclamations/03044.html
Fourth, the Department of Defense even takes this down several more levels:
DoD Instruction 1005.06 March 27, 2008
SUBJECT: Display of the National Flag at Half-Staff
…
- POLICY
It is DoD policy that the national flag shall be flown at half-staff:
…
(7) When so directed by the President or the Secretary of Defense.
…
d. On buildings, grounds, or naval vessels under the jurisdiction of the Secretaries of the Military Departments on occasions, other than those specified in paragraphs 3a and 3b of this instruction, that the Secretaries of the Military Departments consider proper and appropriate. The Secretaries of the Military Departments may delegate their authority, in writing, to authorize flying the flag at half-staff to individual post, station, detachment, ship or installation commanders under their jurisdiction for occasions they consider proper and appropriate; commanders may not further re-delegate this authority
http://www.esd.whs.mil/Portals/54/Documents/DD/issuances/dodi/100506p.pdf
I’m sorry that the NFF got their knickers in a twist over Janet Reno flying the flag at half-staff on Justice Department buildings, but she was quite within her legal authority to do so.
In short, the president can direct that the national flag be flown at half-staff on all government buildings and grounds by law; governors of states or territories and the mayor of the District of Columbia can direct that the national flag be flown at half-staff on all government buildings and grounds in their state, territory, or the District by law; and heads of departments and agencies of the federal government can direct that the national flag be flown at half-staff on buildings, grounds, or naval vessels under their jurisdiction by presidential proclamation. In the DoD, this authorization is extended down to the commanders of individual bases or ships.
In order to refute the argument you need to show that the “Flag Code” says that “only the president” can authorize the flag to be flown at half-staff. This you cannot do since the very Flag Code that you cite clearly says that governors can in their states. You completely ignore the fact that this authority was extended by presidential proclamation to heads of government departments and agencies.