I've put literally miles of privacy fence up over the last 40+ years. We drill either 8" or 10" holes, 3' deep, and set the posts in concrete. I've cut through roots as big as 7" in diameter, and as many as 4 or 5 in a single hole that were over 2" diameter (anything smaller I can chop right through with a sharpened drain spade) but I've worn out several reciprocating saws and hundreds of blades doing it. Never once, in all those years, did I manage to kill a single tree or shrub. No partial die offs, sudden changes in health, nothing. You can take that with a grain of salt, and I'm sure that dire warnings will ensue... but I'm telling you my personal experiences with cutting through roots in those size holes, at that depth. When planting landscape trees and shrubs, I try to just remove the dirt around the larger roots and plant the new one in there with them, or just move the new planting over a little and try again. I've had to move the hole 3 or 4 times trying to hit a spot where I do the least damage to existing roots, but believe me... there's a point where I plop them in the ground and cut any roots that are in the way. I have had new plantings that didn't do well, but again, never killed its neighboring/existing plants or trees.
It is my personal opinion that if you're seeing that many roots, the odds of their owners having a weak root system except in that very spot are astronomically slim. You'd have to cut an awful lot of them to kill anything.
Having said that, I wouldn't cut everything in the hole. Just cut them if they are stopping you from getting the hole deep enough, or from getting the new tree in there. Big ones, try digging the hole away from them if you can fudge the final planting spot a bit. Basically, enlarge the hole in some direction where you don't have to cut any more than necessary.