Lost Judicial Powers and Checks and Balances
First loss would be the items for which the Supreme Court has Original Jurisdiction, meaning it and only it has purview to review
--From Article 3 of the US Constitution
Clause 2. In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be a Party, the Supreme Court shall have original Jurisdiction. In all other Cases before mentioned, the Supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.
A longer description of both the case law and interpretation of original jurisdiction can he found here
Second loss would be Judicial Review which is both a loss of power, and a loss of a check on the powers of both the executive and legislative authority. From Federalist No 78:
If it be said that the legislative body are themselves the constitutional judges of their own powers, and that the construction they put upon them is conclusive upon the other departments, it may be answered, that this cannot be the natural presumption, where it is not to be collected from any particular provisions in the Constitution. It is not otherwise to be supposed, that the Constitution could intend to enable the representatives of the people to substitute their will to that of their constituents. It is far more rational to suppose, that the courts were designed to be an intermediate body between the people and the legislature, in order, among other things, to keep the latter within the limits assigned to their authority.
The Executive not only dispenses the honors, but holds the sword of the community. The legislature not only commands the purse, but prescribes the rules by which the duties and rights of every citizen are to be regulated. The judiciary, on the contrary, has no influence over either the sword or the purse; no direction either of the strength or of the wealth of the society; and can take no active resolution whatever. It may truly be said to have neither FORCE nor WILL, but merely judgment; and must ultimately depend upon the aid of the executive arm even for the efficacy of its judgments.
The interpretation of the laws is the proper and peculiar province of the courts. A constitution is, in fact, and must be regarded by the judges, as a fundamental law. It therefore belongs to them to ascertain its meaning, as well as the meaning of any particular act proceeding from the legislative body.
[W]here the will of the legislature, declared in its statutes, stands in opposition to that of the people, declared in the Constitution, the judges ought to be governed by the latter rather than the former. They ought to regulate their decisions by the fundamental laws, rather than by those which are not fundamental. . . .
[W]henever a particular statute contravenes the Constitution, it will be the duty of the judicial tribunals to adhere to the latter and disregard the former.
Finally, a loss would be able to rule on issues that arise from disputes in multiple districts. Although lower courts would have some review, inconsistent findings in multiple jurisdictions would not have unified case law, as under today's USSC rulings, and therefore both the legislature and executive branches could play off this unequal footing and seek to game the courts
Nominating Lower Court Justices
Both the Federal District Court Judges Wiki, and the Federal Appellate Court Justices Wiki are considered Article III Judges, even though the District judges were created by Congress, as opposed to the Constitution. They are appointed by the President and confirmed by the Senate for life terms.