I'm assuming that is meant as tongue-in-cheek, since it's such a broad statement that it's very nearly proof of its own falsehood. Pretty much everyone overstates cases from time to time! I certainly do!
Feel free to ask any questions if you want to know where any numbers came from
Ok, I'm not able to quite replicate your numbers, but I come close in the end, but with a lower number before precision attack is factored in, and a greater impact of precision attack. Let me know if the following is close to how you are calculating things.
Here's my reasoning, walking through AC 16 which at level 5 was the reference AC I was using (since it corresponds to a baseline to-hit of 60% assuming +4 in the attack stat by level 5).
The fighter archer with 16 DEX should have a base attack bonus of +3+3+2 (proficiency + DEX + archery style), which becomes +3 when using the -5/+10. So to hit AC 16 they will need to roll a natural 13, which is a 40% to-hit chance, not 55%.
So before factoring in precision attack and action surge, they have DPR of 3 * (0.40 * 16.5 + 0.05 * 3.5) = 20.3, not 27.8.
Over 20 rounds, assuming two short rests and thus three action surges, they will have effectively 23 rounds of this, for a total of 23 * 20.3 = 467.
I'm not sure what your intended threshold for a "near miss" is to trigger a use of precision attack, but supposing 20 attacks per short rest, if rolls are perfectly uniformly distributed (which they aren't, but fine), if you use a die any time you're within 4 of a hit (so that it has a better than 50% chance of making a difference), that should happen on average about 4 times per short rest, which is exactly the number of superiority dice you have. Ok, so say you're within 1 once, within 2 once, within 3 once, and within 4 once (since each is equally likely), then you turn about 3.25 misses into hits per short rest, or 9.75 per day, for a total of 9.75 * 16.5 = 161 extra damage from precision attack (not bad!), assuming a picture-perfect distribution of turns and rests throughout the day. So that takes us to 628 total damage, or 31.4 per round. Somehow that's really close to your bottom line, so something must have cancelled out somewhere?
The rogue, meanwhile, vs 16 AC has a 93.6% chance to hit (not 90.9%), because elven accuracy boosts their DEX to 18, so they're working with a base 60% chance vs the fighter's 55%. Their damage on a hit is 19, not 18. So they're doing 0.936 * 19 + 0.1426 * 15 = 19.9 DPR.
So the battlemaster archer, when everything is included, comes out about 58% ahead against a reference AC; more against lower AC, less against higher AC. (Note that they're probably optimized in the wrong direction re: difficulty of opponent; all else equal you'd like to be doing relatively better against tougher enemies, since those are the ones where DPR matters more, though I recognize that AC is an imperfect reflection of encounter difficulty)
So yes, fine, the best fighter archetype meticulously optimized and using their resources very efficiently can do significantly more damage than a baseline rogue when both are using their favorite weapon from a distance*. That's not a surprise. But it doesn't seem to establish "rogues suck at damage in tier 2". If doing 5/8 the damage of the carefully optimized battlemaster archer with the investment of half a non-weapon-specific feat constitutes "sucking at damage" then how many other classes "suck at damage"? Do agonizing blast + hex warlocks "suck at damage"? They're doing less than the rogue with 2 attacks at 60% to-hit and 5% to-crit, with 13 damage per hit and an extra 9 per crit, for a total of 16.5 eDPR.
*Here's something to think about: what happens if one of the other party members knocks a key foe prone? The rogue can do exactly the same damage if they run up and pull out a rapier (using their bonus action to dash if needed). But the battlemaster has to choose between (a) switching targets, (b) attacking at disadvantage, or (c) also running up and pulling out a rapier (if they can even reach). Being less feat-dependent than the fighter, the rogue is more flexible and adaptable, which is one of its main strengths.