I don't give a fig about what it DOESN'T exclude! What I do care about intensely is what it specifically permits!
All it permits is the state legislature - in its capacity as such - to set the election times, places, and manner. A state legislature can only act in conformity with the organic law that brought it into existence, and if that organic law includes review of legislative actions by a state court, then that review applies with equal force to the acts of a state legislature in carrying out its obligations under this provision. The provision did not give to a state legislature any additional power or authority to act that it did not already otherwise have; it simply commandeered that pre-existing authority and, subject to all of the limitations and restrictions imposed on that authority by the organic law that constituted the legislature, allowed the legislature to set time, place and manner.
In order to achieve the opposite result, there would have to be express language that denied review by a state court that was otherwise empowered to review the legislature's actions, and that is not in this provision.
The only way you get to your position is by imposing a penumbra that is not there in order to achieve your desired end-result.
Since it is generally liberals who engage in end-results oriented analysis of the Constitution, that can only make you a liberal.
Bottom line is this: you don't care about the Constitution, you only care about mangling its language to accomplish your personally preferred political goals. That makes you a liberal.