The Founders had assigned the Senate the responsibility of representing the states as sovereign entities. And that among independent and sovereign States, bound together by a simple league, the parties. however unequal in size, ought to have an equal share in the common councils.
It was among a people thoroughly incorporated into one nation that every district ought to have a proportional share in the government.
Yes. Senators were intended to be the advocates of the governments of their respective states.
The 17th amendment made them no different than the members of the House of Representatives, no longer a check or a balance against the winds of short term popular political fevers.