Every discussion of syringes turns into a discussion of gauge — how thin, how much it hurts. Meanwhile the number that actually causes errors is the barrel volume, and it gets picked almost at random.
If your measurement lands at 4 units and you are reading it on a 1 mL barrel, you are reading a mark at the very bottom of the scale where the graduations are tightest, the meniscus is proportionally largest, and a small parallax error is a large percentage error.
The same measurement in a 0.3 mL barrel sits in the readable middle.
Yes. And the error is proportional, which is what makes it worse than it sounds. Being one graduation out at the bottom of a big barrel is a much larger fraction of a small dose than the same absolute error in the middle of a small one.
Coring. Pushing a needle through a rubber stopper can punch out a small plug of rubber. Repeated stabs through the same spot make it more likely, and the fragment ends up in the vial. A larger needle for drawing up cores less, and you swap to a fine one to inject.
Bluntness. A needle that has been through a stopper is measurably blunter than one that has not. Injecting with a blunt needle hurts more and does more tissue damage. The stopper is much tougher than skin.
Not everybody bothers. It is not free — an extra needle and an extra step — but that is the reasoning.
And do not put needles in household waste. A sharps bin is cheap and most councils and pharmacies will take a full one. The person who empties your bin did not consent to any of this.