The modern Supreme Court has expressed increasing hostility to this sort of structure, since the executive branch is supposed to be under the control of the President, and having an agency head who can't easily be removed by the President raises separation-of-powers concerns. And in fact the Supreme Court has recently held that the structure of several similar agencies was unconstitutional. There were thoughts that the FHFA would escape that, because those other agencies had more restrictive grounds for removal -- things like "for inefficiency, neglect of duty, or malfeasance in office". But the Supreme Court just held, in a largely-unanimous opinion, that even "for cause" was too much of a restriction.
To me, the most interesting part of the decision was a rather explosive footnote:
This calls to mind a very similar footnote in Blakely, the case that held Washington's sentencing guidelines unconstitutional: "The FederalAmicus warns that if the Court holds that the Recovery Act’s removal restriction violates the Constitution, the decision will “call into question many other aspects of the Federal Government.” [...] Amicus points to the Social Security Administration, the Office of Special Counsel, the Comptroller, “multi-member agencies for which the chair is nominated by the President and confirmed by the Senate to a fixed term,” and the Civil Service. Id., at 48 (emphasis deleted). None of these agencies is before us, and we do not comment on the constitutionality of any removal restriction that applies to their officers.
Guidelines are not before us, and we express no opinion on them." Shortly thereafter, in Apprendi, the Court did hold the Federal Guidelines unconstitutional, which turned out to have, er, rather dramatic effects.
I would wager that any number of lawyers representing folks with business before the SSA, the Comptroller, and various other federal agencies are more than willing to tee up cases to give the Court the opportunity to "comment on the constitutionality" of those agencies.