You’re missing an important point that means it’s a fine match up. The Manticore is AC 14, +5 to hit and up in the air.
The fighter is likely AC 18 - 22 and can take cover if they want to. As well as being probably having better to hit rolls.
Wow, a 5th level fighter with 22 AC? How did you pull that off? I mean, with a longbow you can't use a shield, so that's, +4 Plate Armor? You must be REAL generous with magical items. Especially since that doesn't even exist in the game.
But sure, maybe they have an 18 AC, plate armor is possible by 5th level. Of course, kind of weird you don't think the manticore can take cover. They live in mountainous areas, and, you know, rocks and trees and clouds are all very likely up there. Manticore could easily attack a fighter who is in a ravine, and fly up and over the sides of the ravine, getting cover, and maybe even total cover from the fighter, every other turn at least.
Oh, and to have better than a +5 to hit by 5th level, you need a 16 Dex. Now, for a dedicated dexterity figther (who likely won't have the strength to wear that armor that's giving them that 18 AC) that's likely. For a strength based fighter? Likely not. They likely only have a 12 or 14 Dex. Meaning they are the same or worse. I mean, they could end up wearing the armor and reducing their speed... but that is going to make running from cover to cover difficult for them.
Also, you are missing that the Manticore has about 20 more hp (at a minimum) than the fighter, even if the manticore is hit more often, it takes more time for them to be killed.
In fact, let's run some of those numbers, hmmm?
We'll assume both parties use tactical play for cover in equal amounts, and just call that a null. So, Manticore vs two different fighters. One is a longbow archer, with platemail, the other is a melee fighter with platemail. Let's see how this averages out.
Fighter Hp with 16 Con: 13+9x4 = 49
Manticore average Hp: 68
Manticore attacks: 7x3 = 21 x 0.4 = 8.4
7x3 = 21 x 0.05 = 1.05
Averages 9.45 per turn and that equals 5.18 turns til Fighter is dead on average.
Longbow Fighter with 18 Dex and fighting style
8.5 x 2 = 17 x 0.75 = 12.75
8.5 x 2 = 17 x 0.05 = 0.85
Averages out to 13.6 per turn, and that is 5 turns EXACTLY until the manticore is dead.
Probably not worth it, but let's try that strength fighter who has a 14 Dex (btw, this is 18 str, 16 con, 14 dex)
6.5 x 2 = 13 x 0.6 = 7.8
6.5 x 2 = 13 x 0.05 = 0.65
Average of 8.45 per turn, which is 8.05 turns to kill the manticore.
So, man, I really hope that cover works for the fighter, because their BEST CASE at 5th level in a 1 v 1 with a manticore ends up with them barely killing it the turn it would kill them. Even assuming the fighter kills it turn 5 and the manticore only gets 4 turns, on average, the manticore will rip through 37.8 of the fighters 49 hp, leaving them with, approximately 11 hp left. From a single fight, against a CR 3 creature, at level 5. And if they are not a dexterity based fighter, with a longbow, and the archery fighting style? Then they lose.
So, again, "the attack is ONLY as strong as a longbow attack" REALLY misses the mark on how these fights can actually go for the fighter.