Why the assumption of left or right, when at least
  in the US most people identify as independents?

Do you wish to cite a source for this or do you wish to gaslight here?

  Voters declaring themselves as independent of a major
  political party rose nationally by 6 percentage points
  to 28 percent of the U.S. electorate between 2000 and 
  last year’s congressional midterm contest in states 
  with party registration, analyst Rhodes Cook wrote 
  in Sabato’s Crystal Ball political newsletter at the
  University of Virginia Center for Politics.

  But as few as 19 percent of Americans identifying as
  independent truly have no leaning to a party, according
  to the Pew Research Center.[1]

[1]

'Difference-maker' independent voters in U.S. presidential election crosshairs

https://www.reuters.com/article/us-usa-election-deciders-ind...