You almost have it right. You forgot the authority that also governs Congress, the Constitution.
...However, state courts have no right to interpret state laws?
With regard to the election of Congress and the Electoral College, state courts have only as much latitude as given to it by the state legislature, Congress and the Constitution.
The reason, and the only reason, that the legislature was mentioned in Article I was that it was the responsibility of said state legislature to appoint the Senators from their states. But with our blossoming Democracy and the final nail in the coffin of our Constitutional Republic in the early 20th century, the 17th Amendment put the power into the hands of the uneducated, vote by their hearts and pockets, and not with their mind public. You can't have it both ways. Either allow the states to nominate Senators through the legislature or all gloves are off and both branches at the state level have the capability of changing it to their whims as well