And see, this is where I get in the weeds.
It seems like you're mostly relying on her on-hit true damage from W if you're not building much AD. But if that's the case, I don't understand why it would be better to go 21/9/0 when it means you're so much squishier and you get less benefit from scaling damage masteries than you would from tank masteries (since you're building mostly tank).
Trinity + Sterak's seem like they have great natural synergy since TF procs would be for 250% base AD instead of just 200% base AD, 300% base AD if the Sterak's bonus procs. Plus it gives you a lot of durability for pretty cheap.
I also don't quite understand the Zephyr slot. If you're already building Merc treads, the tenacity passives are unique. Even with just the AP from Trinity Force, it seems like Nashor's would be the better buy for that slot. It gives you 20% CDR, almost as much ASpd as Zephyr or Wit's end, about 35 on-hit damage, plus the extra 160 damage and 40 healing the from the AP ratio on your ulti, and an additional 40 damage on your E. Since Q applies on-hit effects, you're actually doing more damage by applying a Nashor's on-hit with Q (and AAs) than you would get from the 25 AD of Zephyr. So in a single ability rotation, Nashor's gives you a net 150 damage over Zephyr and an additional 10-ish damage per AA; the trade-off is 10% ASpd for 10% CDR and a 150g additional cost.
As for BC, I guess I don't really understand how Irelia can get by without any ARpen if she can't pull off a win by the end of mid game. Seems like it would be pretty easy to get stacks between Q, R, and AAs, and a few stacks on engage would also increase the healing from R pretty substantially. I get that the pure AD isn't spectacular for her, but the CDR is great, Phage is necessary, and the aoe armor shred from her ulti seems very strong.
On another separate point, I've been wondering whether it's viable to take Challenging Smite and building Devourer even in top lane. It's super cheap (and gold efficient) and has very high aspd with on-hit damage that will scale into late game, potentially giving you a second win. A snowball on your snowball, if you will. Plus the on-hit damage and incoming damage reduction from Skirmisher's seems like a guaranteed trade win. Why doesn't it work?
EDIT: FYI, I'm not arguing with you in the sense that I think I know better. I clearly do not. However, the hybrid scalings and on-hit make it very difficult to understand how and why certain items are best on Irelia compared to others. Like I said before, her power feels very opaque.