These are fair points .. but not especially so for objects this distant.

Given what you know from your interest already you might like to, say, work out at what distance objects appear to essentially fixed in position when viewed from either end of a 2 AU baseline.

Having a background wallpaper of 'fixed' objects that can be occluded helps to inform us about the closer occluding objects .. but triangulation doesn't assist in depth sorting all the distant objects that appear fixed.

My comment above assumed (perhaps incorrectly) that you were referring to the title topic and implying that our motion about the sun was enough to determine the distance to these far objects as the JWT moved through a few orbits.