800k working-class Americans don't get paid when the government shut down. Do the US politicians still get paid? If yes, why so?
Article 1 Section 8 Clause 1 of the COTUS states that Senators and Representatives shall receive "compensation" for their "services." This is "ascertained by law."
So they must always be compensated so long as they are elected by their state or district, and they are.
I suppose one could attempt to change the law, but I think this is fruitless. If they can't agree to one law that is supposedly desirable, why should we think asking them to change another law would help them change the aforementioned law? After all, they are the ones with the power of the pen at the moment. By the time you have any say, I would like to think this problem would be behind us. Remember, they will always be constitutionally guaranteed "compensation," and the courts will never let that mean something absurd like "1 Penny per 1000 years."
As to the incentives benefit you hypothesized:
I do not think this is necessarily something that would have more than a negligible benefit. A great deal of politicians are already so rich that the salary they receive from congress is basically lunch money for them. For anyone else, it may be an incentive...but you don't know what kind of incentive yet.
It could cause the best guys to leave because he cares more about his child getting through Harvard. I wouldn't blame a guy like that for leaving. Any talk of "greater good" here is hogwash.
If it were me, I would arrange for a public mansion to be built for each of them and provide servants and everything. But not one red cent worth of money or any other material item that he could keep past his term.
But everyone has their whack job idea right?
Yes, members of Congress are paid salaries when the government is shut down. They're still paid because all employees of the government that weren't furloughed (politicians or otherwise), are paid their salaries. Some members of Congress have stated that they will forgo their salaries while the government is shut down, but they're still legally entitled to them.
As of now, it's unknown if employees who were furloughed will be paid retroactively.