Here is a post by /u/Alt_Mayday from a couple of months ago that shows how inaccurate most iron sights were while moving the mouse:
The recent patch notes included:
Update to Ironsight Accuracy
Apex guns move around subtly while ADS — they rotate and sway as you move your aim, they bob up and down and side to side while moving, etc. We have special tech that makes sure the reticle on the optics stays centered even as the gun moves around a bit. However, until season 4, the ironsights on weapons didn't have this tech. That meant that the ironsights reticles could move around with the weapon and actually be slightly inaccurate — they wouldn't point at the screen center where the next shot would fire. With season 4, we have updated most of the weapons' ironsights to properly stay centered now. Not all ironsights are upgraded yet, but we will be updating the remaining weapons as this season progresses. The weapons that will be updated for Season 4 are:
Longbow DMR
TripleTake
Charge Rifle
Sentinel Sniper Rifle
Havoc
R-301
Flatline
Spitfire LMG
R-99
Alternator
Mozambique
Mastiff
RE-45
Wingman
I noticed this especially with the Mastiff since I almost always had scopes for the other weapons, but never for the Mastiff for obvious reasons. Try it out in the firing range, the iron sight is very centered and it feels way more consistent now.
Source: https://www.reddit.com/r/apexlegends/comments/ezcxm6/have_you_ever_wondered_why_the_mastiff_or/