Why exactly is on-hit "dead"?

1. I don't think it's dead, I just think it's weaker than traditional Crit/AD/AS right now, making it highly situational.
2. It's weak because it provides no sustainability compared to traditional AD. You can't spell vamp your on hit effects, and life steal provides less healing if you went On-hit rather than AD.
3. It's also a strategic flaw. You do less damage to their dangerous people (carries) and more damage to their less dangerous teammates (tanks/supports).

In order for on-hit to be effective, it would have to reliably dish out more damage than AD. Currently, that only really works against HP stackers because the big ticket On-hit item is Madred's Bloodrazor.

What this comes down to is that you need to design a team around on-hit champs because they excel at the poke game...they're able to shred up the tank really quickly so if your team pokes hard enough the tank will be so weak that when the engage comes in, your team will melt the tank before the tank can get all of their disruption in. That's the hope at least. There's really no reason to play On-Hit if you're going after carries.

It can't keep pace in the current meta where soak-oriented characters gear for sustain.

On-hit items aren't good enough. You get more DPS for cost going straight AD.

The only time I see it as being useful, is if your team has some other AD, and you expect your opponents to stack a lot of armor, like if they have a Rammus.

It's always been lower DPS than IE+PD+LW. The reason to build on-hit is because it's stronger at lower gold levels than carry builds, results in higher AS (for character on-hit procs), and provides nice side benefits (MR, magic shred, and now HP).

Wit's End is still a very strong item, just not for the AD carries.

Vayne says on hit is not dead.

Depends on the champion really. Going on hit on someone like trist or graves or anyone without a kit oriented reason for going on hit is not wise for the above stated reasons. That being said if you are playing a teemo a kog or a shyv/skarner/xin where you have a compelling reason to stack AS then it is by no means a weak or even mediocre build.

For example Kogmaw is probably considered to be the best onhit champion due to his natural permanent AS boost and the %hp damage on his w. Malady, wits end, madreds into gunblade into late game trinity is a horrifily powerful build. It mighgt be incrementally slower at killing carrys and poorer at dueling but that will be an academic debate most of the time as the ability to kill their initiator, then kill their bruiser as they come into range but before they have a chance to really do anything allows your assassins to stick to the carry without being peeled. But as before notednit requires a certain built in preference for the kit.

You are playing Vayne wrong.

She has one (bad) skill that scales with attack speed, and then... AMAZING scaling with AD across 3 skills.

Let's look at the champions who are considered good for on-hit build:

Kog'Maw: Does better damage with AD, isn't expected to do anything early game, is safe so doesn't need survivability.
Kayle: Does better damage with AD, isn't expected to do anything early game, is safe so doesn't need survivability.
Teemo: Solo top, needs better sustain and survivability to live up there. Focused on kiting. Build Atmogs.

There's also the problem of Atmogs does similar damage, keeps you alive better, doesn't have the penetration problems, and actually lets you dedicate more slots to straight damage late game.