Why didn't Washington sue Trump directly in the Supreme Court?

Question

Article 3 of the Constitution provides:

In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the 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.

So far as I can tell, the Eleventh Amendment does not modify this in a way that makes a difference in this particular case.

Washington is a state, and it is party to Washington v. Trump. Why is the lawsuit currently before a district court instead of SCOTUS? Does Washington have the power to decide the venue? If not, could the Supreme Court invalidate any rulings of the lower courts as lacking jurisdiction?

Answer 1

Original jurisdiction is not necessarily exclusive jurisdiction.

[T]he original jurisdiction of the Supreme Court, in cases where a State is a party, refers to those cases in which, according to the grant of power made in the preceding clause, jurisdiction might be exercised in consequence of the character of the party, and an original suit might be instituted in any of the federal Courts, not to those cases in which an original suit might not be instituted in a federal Court.

Cohens v. Virginia, 19 U.S. 264 (1821)

However, "in the Judiciary Act of 1789, Congress made the Supreme Court’s original jurisdiction exclusive in suits between two or more states [...]. The Supreme Court’s jurisdiction over the remainder of suits to which a state was a party was to be concurrent." (Federal Judicial Centre - History of the Federal Judiciary)

Answer 2

To elaborate on K-C's answer somewhat, suppose we understood

and those in which a State shall be Party, the supreme Court shall have original Jurisdiction

to imply exclusive jurisdiction. That would mean the only cases that could be tried outside of the Supreme Court would be cases between people, or involving municipalities (or territories). Crimes are actions where a state is one party and an accused is the other. If cases involving a state, such as theft, murder, or suits by a state over fraud had to be tried in Supreme Court, that would bring the Supreme Court to a halt, which would be an absurd result.