Erm.... Okay. Let's take a look.
I listed the sort of things I was thinking in the post above. I didn't list them originally as I didn't think we were going to get down to minutia as I thought this was a pretty hand wavy kind of discussion. But yes, I did assume some basic magic items for an epic level character. This might not be appropriate for some super low magic item settings I guess.
I also wouldn't expect an entire army to be fitted out with chain & shield (that's very expensive) but again, I suppose it depends on the setting. Yours could be extraordinarily well supplied.
Most armies also don't call down ranged attacks on their own troops location just to try to hit one guy standing in the middle of hundreds of their own guys..... Again, I suppose your campaign setting could be unusual in this regard. They might be very bloodthirsty.
But honestly, we can tailor the example however we like, but given the infinite possibilities of campaign settings, character builds, and actual player-DM storytelling interactions, to me there is plenty of room for this legendary swordsman to be able to acquit himself well in this scenario (or versions of it).
What is your fundamental disagreement with this?
I think part of the issue is that 5e changes the dialogue between a high level character and a swarm of mooks. In older systems, you could expect the high level fights to mow through mooks like grass, but scaling wise that is not how it works in 5e. Now that's true for casters as well, for example its harder for a high level wizard to be "invincible" versus a mook mob like they were before, but considering their arsenal of area effects they certain have more tools at their disposal.
Just to throw in a little more fun math, I will try a more "mook" scenario, and give the fighter some actual magic items, just to see how things would go.
20th level Battlemaster Fighter
20 strength
16 con
HP: 210 (I just rolled second wind in for convenience)
AC: 22 (does not take critical hits)
Attack: +13 x4 (x8 in first two rounds)
Damage: 1d8 + 10
+3 longsword
Adamantine Armor
+3 shield
Dueling Combat Style (so +2 damage)
AC style (+1 AC)
"Mook Archer"
HP: 8
AC: 10
Attack: +2 (proficiency only)
Damage: 1d8
So these are guys with 10s in all stats, basic proficiency, but otherwise nothing special at all.
Scenario 1: Straight up
These stats actually make running the scenario pretty easy, fighter is going to kill a mook every time he hits, no crits to worry about and don't really need to think about maneuvers. We assume the archers are close enough in formation that the fighter can get to any mook he needs to, so its really about how fast the fighter can kill the mook before he is taken out himself.
Result: 20th level Fighter can take on 89 archers. 90 will result in their death.
Scenario 2: Swap Adamantine armor for +3 armor
Lets try a different magic item that is still very powerful, but not really that helpful to the fighter in this scenario. So now when the mooks hit, they are actually critting. Let's see how this effects the result.
Result: 20th level Fighter can take on 64 archers. 65 will result in their death.
Scenario 3: The most mook of mooks
So lets get really schlub. These guys have clubs, attack in melee, and don't let their buddies move in and get attacks (aka they are complete idiots). So they get 8 attacks a round, though they are crits.
Result: 20th level Fighter can take on 410 melee mooks.
Scenario 4: So mook you can't believe it
You thought the last scenario these guys were terrible. But ok, no weapons, no armor, just guys using their fist! (aka 1 damage a round, and since its not a die its not crit multiplied). And still 8 guys in melee a round. Lets see what happens!
Result: 20th level Fighter can take on 2005 melee mooks.