I'll agree with you on the CDR (except that it's almost becoming a worthless stat because so many items use it now that it's moot point to "build CDR"), and the increased AS does passively let them deal increased damage because all of them scale better with AD than AP (except Corki, but even he has reliability on Crit Chance).
Jhin was great with it because of his passive, and due to Spellblade you could deal even more damage per auto if you Q, W, or E before the attack. Yi could abuse it because of his Q, E, and R, and could mow down opponents easier thanks to IE and Shiv. Pantheon gets greater burst damage because of how well his abilities scale with AD; Vayne...I'll give you Vayne, it's more because of Rage and Spellblade that it worked for her but due to its cost there were better options.
Mainly, it was due to the stats and Crit Chance that it was such an Early Game staple, as well as its Rage and Spellblade Passives making it easier to have a more successful Early/Mid Game. Now that its Crit Chance is gone, though, they have to look for others. And due to the extremely limited amount of items that actually give Crit Chance for Melees, they're forced to be extremely predictable if they want to get anywhere with it.