You could hit 1000 HP at level 6 (with an inept DM).
The real issue is the semi bloated HP along with the 6-8 encounter per day game design.
A DM who uses a lot of high CR enemies has tons of HP. The DMG has you run through an expected 900-1200 HP a day if you follow the chart at level 6. If you fight 6 cyclopsi, you are burning through 828 HP. Even if you go with lower CR monsters and throw in trash, the HP per encounter goes from 75-150 HP 6 to 8 times.
Once you realize this as a play, it nudges your damage dealers to want as much damage as possible. And if your tanks rely on scores, it nudges them as well.
It isn't required. However after level 6 and 7, you might really feel the need to buff your primary if you didn't start with a 16 nor buff it by then.
It doesn't really matter how many HP or how many encounters. 20% more damage* means that, by yourself, you will on average kill a monster in 4 rounds instead of 5. Because of fenceposting that will happen against some monsters more frequently than others. Or, if you're fighting monsters that die twice as fast as that, 50% of the time you will deprive the monster of its last action.
The thing to remember is that it's not just the straight +1 damage; it's also the increased chance of hitting, which is computed as the relative change. That is, if you had a 50% chance to hit, and now you have a 55% chance, you hit 10% more often. Which means 1/10 attacks you get a full extra hit in. If you previously had a 25% chance to hit, you get an extra attack 1/5 times.
This effect does not only appear over large numbers. That's nonsense. It
converges on the predicted value over large numbers. Which means that, over small numbers (say, one adventuring day) half of us will see a diminished effect, and half of us will see a greater effect. (Do you feel lucky, punk?)
Now, the value one places on that is subjective. Some people would rather have pointy ears, and that's cool. But that's the math.
*Again, this effect increases when the AC goes up If you're unlucky enough to encounter an AC 19 monster at level 1, the fighter with 16 strength does 35% more damage than the one with 15. If you also have disadvantage it's 60%. If the target also has partial cover it's 80%. Etc.