I think part of the issue with discussions like these is that players who oppose the above statement tend to reason in terms of raw power: because AP champions in general are balanced to win rates around 50% on average, with okay play rates, their itemization must be fine, so goes the argument. The problem with this is that it focuses on the wrong parts: sure, AP champions have been balanced over time around their itemization, as is literally any other champion, but that still doesn't prevent the fact that AP options are far less diverse, interesting, and overall powerful than AD options.
In recent times, we've experienced many major reworks to certain classes, which also featured updates to their items. Marksmen and mages both received two item updates in the past few years, but then when we look at assassins and juggernauts, only one half got their itemization looked at. AD juggernauts got items like Sterak's Gage and Titanic Hydra, and AD assassins received Lethality, but AP juggernauts and assassins got squat. In fact, over time their options have worsened, as Rylai's, once a major component to AP juggernaut builds, got nerfed, whereas AP penetration outside of boots continues to be dampened by health, and railroaded into a single item purchase.
A common counter-argument to this is that AP and AD are designed differently, which is true, but the fact remains that there is a clear difference in the amount of attention given to AD and AP items. Moreover, what were once significant differences between AD and AP have blurred, even vanished over time: Zhonya's used to be the pinnacle of AP itemization, now any champion can access its effects with a 600 gold purchase. AP items in general were renowned for providing large amounts of utility, as opposed to AD items mostly just offering damage, now even items that seem almost purely oriented towards raw damage-dealers have some utility thrown in for good measure (e.g. Duskblade). Cooldown reduction, which once only mages and supports could access in meaningful amounts, is now available to all, to the point where it's no longer difficult to cap out, regardless of class. AP items used to grant major burst, but have been neutered over time, with AP champions turning towards DPS in the process, while AD options have been piling on increasing amounts of front-loaded, unavoidable burst via on-hit effects. On top of all this, even the rune system carries a bias, as there are no real defensive options against AD champions, and the one Keystone that could have helped out AP top laners was excluded from giving adaptive stats, under the pretense that some AP-scaling champions would abuse it. In the meantime, the Keystone is so heavily abused by all sorts of top laners that Riot is considering nerfing it. It really isn't a good state of affairs.
With all this said, it's worth mentioning again that there is a distinct difference between saying AP itemization is weaker than AD, and saying AP champions are weaker than AD. The latter is demonstrably false, but the former is evident. For sure, buffing and diversifying AP itemization would mean bringing appropriate nerfs to AP champions, but honestly, I think that would be preferrable to the current state of AP champions, where many of them have simply been given enough raw stats and numbers to be able to cope with an itemization system that doesn't adequately cater to them. Mages may probably have the best choices right now, but assassins and fighters should not need to pick from their same item pool, particularly since neither of the latter are reliant on mana, a stat that Riot just decided to make core to AP itemization for some reason.
and 
Wow this item has a cd too better hit those skill shots
maxed gives 45 armor
( 
or 
were getting this item + 
against a fed AP when it was in meta
which is also extra burstttttt!!!!
- HP that you forgot