Nope, you could have a Senate appointed by the President.
The senate is still intended to represent the people. xP
In what world do you live in? Please tell me how the Senate even remotely considers the thoughts of it's people. If that were the case then we would never heard about candidates that are say pro-choice or pro-life or against amnesty or for it, because their personal views wouldn't matter since they're supposed to represent their people.
WHAT YOU THINK OR WANT DOES NOT MATTER. IT HAS NOT MATTERED FOR A REAL LONG TIME.
Sure, in the 1800's this might have been okay with the politicians actually being part of their community and not taking trips all over the world or lobbying for money for their re-election rather then taking the time to sit and talk with the people.
Well, a lot of politicians back in the day traveled a lot, Benjamin Franklin was especially useful due to his ability to persuade the French to support us.
You also have to remember that slavery used to exist, and that was you know, not a good thing for the people.
The Senate members need to get elected, so they need to appeal to the people. They represent the people's issues on the matter, and get elected by saying beforehand which issues they'd represent, to make themselves appear favorable to the people. If the people's will had no bearing on if they were elected or not we wouldn't know what their opinion's were on certain things, we wouldn't see campaigns by them in the first place, since it wouldn't matter. That's kind of the anti-thesis of what you're suggesting; it's one thing to say that they're corrupt, it's another to say because they campaign to the people to show they possess the same opinions, so you know you have a politician you can trust, that the people's will doesn't matter.
In a representative democracy, you have to make it clear that you'll represent the people, by and large, which is what they do. Politicians that stick to their principles either step down or try to convince others that their ideas are best to get elected; usually, people want results in addition to representation, so you have to come up with an idea you think will work and then convince everyone it's the right thing to do.