Fun fact: republicans hold a majority in both chambers. The government shut down because republicans won't vote for their own bills.
Cloture is a very valuable tool for the minority in the Senate to keep the majority in check. Cloture is just a rule which can be changed by just a majority vote. I think it's a mistake for either party to do away with this rule. For example, during the Obama Administration, the Democratic members of the Senate changed the rules to allow federal judges to be approved with just majority vote. Then, during the first Trump Administration, the Republican members changed the rules to include supreme court judges, which lead to the current supreme court.