Magic damage doesn't scale as well as Physical damage in most cases, hence MR being about half the quantity of armor on items. Consider how magic damage typically scales - it's the base value of an ability + an ability power ratio. For rank 1-5 abilities you typically need 150-400 ability power to just double their damage output, which is rather substantial. Ultimates tend to require closer to 700 ability power. And it's not like the base damage of most nukes is that much better than a critical strike, you're looking at ~70-280 with most QWE abilities. To multiply magic-damage output you need to grab magic penetration and cooldown reduction, neither of which can double your DPS by themselves due to it's rarity/cap - but with a combined assortment of both you likely can.
With attack damage you can stack them linearly with more attack damage, needing only 50-110 to deal double damage, and you can double your DPS with critical strikes, attack speed, and armor penetration - and cooldown reduction helps as much here in many cases as well.
With attribute scaling, you need to keep up with an ability that can go 1->2->4->8, compared to just 1->2, so the percentage you can mitigate with magic resistance doesn't need to be nearly as high as what you need to mitigate with armor.
Also, it's not hard to win handily against AP champions with Maw of Malmortis and Wit's End. Throw in a pair of Mercury Treads and you're rocking close to 150 MR unbuffed - that should allow you to drain-tank the likes of Katarina, even with the healing debuff her ultimate applies.