It was the Gingrich "Contract with America" Congress that balanced the budget and spurred economic growth, primarily with a conservative version of Welfare Reform that put time limits and work requirements on welfare.  Clinton had vetoed that bill twice, but in December 1965, Congress shut down the government, lifted it over Christmas break, and resumed it in January, and held their ground for 57 days before Clinton relented and signed the bill.  It was a tremendous success, and Clinton took credit for it.

Did the shutdown hurt the Republicans at the polls?  No. Bush43 won.  And hardly anyone can recall being hurt by that 57 day shutdown.

Fun fact: During the Carter Adminstration, which had a Democrat Congress, Congress shut down the government on their own president 7 times in four years.