Allow me to wish you good luck first, it'd be really awesome if you could get that internship!
I'm not really an old LoL player (it's been a bit more than a yeah since the day i played my first game), so I didn't really see the Morgana FOTM time or the farm-focus meta. I really like your suggestion, but what Bullet says is true; even if she gets that new proc passive in mid lane, it doesn't allow her to trade well in lane if her opponent dodges the Q.
However, such a new passive is a straight buff support-wise because she doen't use spellvamp at all in a duo lane (or at least it's pretty negligeable). But then you don't build that much AP immediately.
One of the main problem of Morgana (and Kennen to some extend) is her reliance on Zhonya. If you don't get it, you do peanuts in teamfights. And that item is not of the the easiest things to build; in fact you need 1.6k gold for one of its component. So, when you got that cheeky snare and killed your mid lane, you have 1k gold and you know teamfights are coming soon, you still can't spend that gold. Hence some situations where you find your damage pretty weak. But contrary to Kennen, she's immobile if she doesn't have flash, and she needs to land that snare. The camp is too binary in her current state until it comes to large fights.
Thing is, you also have to consider the champion in the current metagame. You don't see Anivia for the same reason as Morgana: pushing and farming from a mid to long range for the sake of gold is not what we're looking for today. Today we clear to kill lanes to kill people. To make plays at their buffs, etc. Roaming to bot as Morgana is rewarding if you can land your Q, but for that you have to be very good with skillshots, or you have to somehow surprise them to get a good shot. So, if you want to farm then roam, there are better champions with slightly weaker farming but better roaming and flexibility, like Zed, Diana.
But if the meta shifts (for some reason) to a more farm focus game, then yes you'll see her, Anivia, and other farming champs more often because she's really good at it while having a huge impact in end game, she's really good at it in her current state.
So, the problem here is, you have to wait end game to USE the gold you got, but even then it's a bit tedious, and you put yourself at risk by making plays (get closer to ult, flash flash flash flash, what if they dodge Q, do I hang on to my W until Q hits something; if Q hits something I don't want to W, do I still use W?). In mid game: yeah OK it's fine but my ult doesn't have the feeling of "omg optimal ult" yet. In early game? Farming didn't have an impact yet.
One thing I'd like to point out with the new passive; the passive will have to be strong enough to reward you if you get a Q, or massive E off. But then you feel even weaker if you don't hit Q because you can't spellvamp of your minion wave anymore, this is especially true versus mobility based champs. Also, if you pull of, say, a 4- man ult, they will have have Marks right? While it should be game winning, you can't rationally proc the Marks yourself with only one W and your auto attacks, so you might have the feeling " well my passive is good, but sometimes i don't get to use it ot its full extend". If you allow teamates to proc it, then it'd be a bit too strong imo, and make Leona players pretty sad.
So yeah, just my two cents, hope I can help you!
N.B: No, don't go for Frozen Heart on her if you have Zhonya already. Not stat efficient if you already have Zhonya. If you don't have Zhonya but you're still going melee range for FH you're either really fed (then you don't go for FH!) or you want to draw attention and get killed immediately, FH won't help you withstand a whole team.