Please provide a link to the official duties of senators that includes ‘representing the people of their state’.

I agree that the US is generally a form of representative democracy, however the senate explicitly represents the states, not the people within the states, and always has. The fact that senators are equally allocated per state, rather than by population, with each state, regardless of population have the same number of senators does not support the theory that it is a representative democratic institution. The intention was explicitly to maintain the power of the individual states within a federation.