Re: Just an idea
Alexandria Ocasio-Cortez famously said she could not afford to rent an aprtment in Washington DC until she got her salary after being elected. So I don't think that is quite 'poor' but it's not rich. It also means that in the event of a shutdown, with the rule proposed here, she could no longer pay her rent and thus would presumably not attend Congress, vote &c.
And this, again, is why this idea (attractive as it is) doesn't fly: rich politicians have little to lose personally from a shutdown while poor ones have much more to lose. So rich politicians can starve out (not literally) poorer ones by imposing shutdowns. That's anti-democratic: poorer people should not be disadvantaged as politicians.