You're underestimating how much power Congress legitimately has over the rest of the government, when they choose to exercise it.
All money spent by the federal government in all three branches, including on judicial clerk compensation salaries, is determined by Congressional authorization and congressional statutes regarding employment and compensation. Congressional statute law heavily regulates hiring throughout the executive branch, despite that being a separate branch of government from Congress. There's no reason the judiciary is any different.
What's more, while the procedural rules of the federal courts are mostly set by the federal courts themselves, this is done under the authority of an act of Congress, which has been amended several times both to increase and reduce the courts' self-regulatory authority. As for the Federal Rules of Evidence, Congress refused to allow the version of those which the Supreme Court originally approved to go into affect, and then they later passed a substantially altered version as an Act of Congress.
The Constitution itself makes explicit Congress's ability to regulate the Supreme Court's appellate jurisdiction (which applies to almost all of its docket) and even to reduce the scope of that jurisdiction. And it has pretty plenary power to add, remove, and restructure the inferior federal courts, aside from whatever tenure protections apply to individual judges who have not been impeached by the House and removed from office by the Senate.
And, as others have said, the Senate even individually approves federal judges as per its constitutional mandate under the appointments clause. This is certainly not a separation of powers violation.