> The administrative law proceeding is a judicial proceeding

It is structured like a judicial proceeding, but it is not a judicial proceeding in the sense that it doesn't exercise the "judicial power." For example, everyone agrees that determining violations of criminal laws is an exercise of judicial power. So the SEC cannot prosecute you in front of an ALJ and impose criminal penalties. SEC v. Jarkesy also held that the SEC cannot impose civil penalties in front of an ALJ: https://www.congress.gov/crs-product/LSB11229.

On the other hand, Congress can create a procedure in an executive agency that looks like a judicial proceeding, so long as it only does what an executive agency could do anyway. In Oil States, the Supreme Court held that the Patent Trial and Appeal board was constitutional, because patents were a "public right" that could be determined by the executive branch: https://www.supremecourt.gov/opinions/17pdf/16-712_87ad.pdf. A Patent Examiner can grant you a patent and can revoke an already-granted patent. That's an executive function. Back in 1790, Thomas Jefferson was doing that himself as Secretary of State. So Congress can create a court-like proceeding in the Patent Office to challenge a patent's validity, and that's fine because the result is something the executive branch could do anyway. It's not an exercise of the judicial power.

> But if the ALJs and the SEC rulemakers are under the direct control of the president, then the president is effectively performing legislation, prosecution, and adjudication all at once... That is precisely the kind of scenario that separation of powers seeks to avert.

Under your interpretation, the SEC is exercising legislative, executive, and adjudicatory powers all at once--regardless of how much supervisory authority the President has. You're correct that separation of powers seeks to avoid that. But the way it does not is not by creating agencies that exercise all those powers independent of the president. It does so by creating three branches of government, and requiring the executive, legislative, and judicial powers to be exercise by the respective branches. The problem you have identified is a problem with administrative agencies exercising legislative and judicial powers, not a problem with Presidential control over the executive branch.