Your first premise is in error. Angus King is NOT an independent, nor is he a centrist. I live in Maine, know King, and have worked with him. I’ll use other forums to berate his standing and spare you the details. In reality Maine’s Senator Susan Collins (and formerly Senator Olympia Snowe) is a true independent and centrist.
We really need more than one additional party. The event that may finally trigger a new way in political maneuvering would be to change Senate rules to eliminate the filibuster. Each of the two major parties have splinter groups and the leadership of each party bends to the whims of the most radical ideas of their sub-groups. Incentivize the splinter groups to form new political parties and have future legislation built with consensus across all elected officials.