And you have the right of it here.
Unusual to see an argument where both participants are correct, but here it is in living colour.
I'm going to explain this in video game terms. Let's calculate your average damage per round.
Let's assume you have a 9% chance to miss a melee attack. So of every potential 1 damage per round, you only do 0.91 or 0.83 of that damage.
If you increase your chance to hit by 1%, this goes up to 0.92 damage per round, which is a 1.1% effective increase.
So if you deal, typically, 1d8+5 per round with a longsword, and you attack twice, we know your potential damage is 19 damage a turn. If a battle last four rounds, your potential damage is 76 (not including crits).
But let's say you have a +9 to hit and you're up against AC 18. You have a 40% chance to miss. That reduces your average damage by 40%, to 11.4.
Increasing your chance to hit and damage by 1 (say in this case, by picking up a magic sword), only increases your chance to hit by 5%, so your average damage is now 65% of 20 per turn, or 13. That's it, you only gain 1.6 damage per turn.
Now there are some other concerns, like if you're a Battlemaster, or your character is Dex-based and you get AC and other things, but the difference between a 16 Strength and a 20 Strength is a 10% chance to hit, and a small boost to damage.
The actual amount of damage dealt will depend on the weapon and how many attacks you make.
But as Maxperson pointed out, if you were already going to hit, you will barely notice the increase to damage. All we're doing is shrinking the chance to miss.
There is a small damage increase- the difference between a greatsword fighter at level 1 with 16 Strength and a bless spell vs. the same fighter with 20 Strength is noticeable, but small, something like 1.05 damage per turn. A pretty small increase either way.