The Court distinguished between executive officers and quasi-legislative or quasi-judicial officers. The Court held that the latter may be removed only with procedures consistent with statutory conditions enacted by Congress, but the former serve at the pleasure of the President and may be removed at his discretion. The Court ruled that the Federal Trade Commission was a quasi-legislative body because it adjudicated cases and promulgated rules. Thus, the President could not fire a member solely for political reasons. Therefore, Humphrey's firing was improper.
https://en.wikipedia.org/wiki/Humphrey%27s_Executor_v._Unite...Sounds like what the parent was saying, so not flatly incorrect.
What the parent said:
“If the law says the President cannot fire someone, or interfere in an agency’s work, then the President cannot.”
This is, indeed, flatly incorrect. Congress cannot pass a law requiring that the Secretary of State or Defense or Treasury be fired only for cause. The SCOTUS case knocking it down would likely be 9-0.
“Congress writes the laws and can make them say whatever they want” totally ignores separation-of-powers concerns that the Constitution and its guardians in Article III courts take very seriously.