Her passive can give her a stupid amount of Tenacity so It's insanely impossible to shut her down if she's fed "Just CC her". Well that's what they said about Yi until he did 900 true damage to me in less than a second with Devourer. On top of that, people build her as a tank. I do see Irelia players go Tri Force > BoRK sometimes, but that's only if they're carrying. Most Irelia players typically build full tank though.
As for her being a Mage. She summons a ton of oversized daggers and throws it at your team to heal herself. Actually recently I saw an AP Irelia carry. I don't think I've ever seen AP Irelia Mid but apparently she one shot our Zed so that was cool.
The thing is that while Jax, Nasus, Aatrox, Yi, Trundle, etc outscale her into late game, if they fall behind they're near useless. Irelia has built in true damage, a point and click slow/stun, and she pretty much is immune to all CC. If Irelia falls behind she's still an extremely high threat to the enemy team unlike Jax who would instantly die in a teamfight without his dodge, or Aatrox who can pretty much be one shot.
I won't argue with you on Nasus though.
hur dur lemme feed 20 kills to the enemy team but because i pressed Q 150 times i can literally 1v5 pentakill in the enemy fountain and one-shot the enemy ADC with a full tank build.