Well I'll try and give you a really in depth answer . I wonder this sort of thing a lot as well, not specifically with Zed(I've never played the champ one time just FYI) but with most champs that seem to have multiple abilities with poor AP scaling. You can see this with some of the Devourer Junglers to be even more specific (shyvana). I see her built either tank or with AS/AD/Lifesteal. I'm pretty sure she has at least 2 abilities that scale with ap, but you don't typically see her built with a rylai's.
However you have confused abilities/spells with the type of damage they deal. SO lets just clear that part up. All champs have abilities, but what defines a champion as AD or AP is the respective AP/AD ratios on the champs most important abilities. This also needs to account for passives. There are only a few champs that have kits with either ALL AD or AP ratios on all of their abilities. SO you wouldn't consider ZED to be an AP champion based on his use of abilities, rather HE would be considered AP if required AP items to do AP damage. SO AP zed would have a build based on AP items.
What separates him from AP Midlaners( I.E viktor) is item build . According to his wikia he has no AP ratios whatsoever. SO he would actually yield very little damage from building AP items. Now he could use some of the stats an AP item would give and have it be useful, but his abilities wouldn't necessarily do much more damage from hitting an enemy. Unless its something that has on hit effects/AA effects. For example Pretty sure Rylai's would still slow, and maybe Liandry's would do damage over time(never tested so try it out and see). However, his abilities would still be doing base damage even with 400 ap since he has no ratios that use AP. Works the exact same way in reverse, however your basic attacks will obviously grow in strength (AD zil or something like that)
Let's look in depth more here for fun. I'll use a champ i know well for my example, Rengar. AP Rengar changes his play style rather dramatically. Rengar uses his abilities in quick succession, and then if need be, likes to reset before reloading the combo again (Sorry if you knew this). Most people would consider him "AD"and his kit has mostly ad ratios (Q,E) , but his W has a very stout ap ratio(70%). So you stack a few items, and get ahead in the game, and you're a jumping kitty with a very deadly roar. However since the rest of his kit is focused around doing AD, u can be in a bad spot if you miss that ability, or screw up the combo for the max burst. So its easier to deal more consistent damage building AD Rengar vs AP Rengar since you only have one source of AP to deal AP damage with. He's still technically an AD champ despite my enjoyment.
I find it strange and intriguing that champs have ap and ad ratios. Especially on new champs like JHIN, who has great ap and ad ratios. So it comes down to a couple things - Personal Preference (play your way, tho your way may not be efficient/easy) - And of course LOL Math - What items will make your champ and its abilities do the most damage and/or have the most utility or a combination of both. Many players use the mathematically correct builds to determine what damage type a champ is, But there's nothing wrong with having fun right?!?!
I hope SOMEONE READS THIS(LOL) and found this useful! I feel the community can be a little to dismissive at times.