If the goal is to add something to the healer's responsibilities, I'd go the way of debuff removal on party members (which doesn't happen enough and is almost never crucial to clearing a fight) and dispelling enemy buffs (which doesn't exist in this game for anyone that's not BLU). Interrupting/Stunning and the like should be absolute last resort for healers, because there should be other roles covering that in a group.
Procs I think should be tied to the healer role rather than being damage-oriented as the OP suggests. The problem is that a good number of the possible procs are already in use (Cure's proc that removes Cure II's MP cost, Benefic's procs that ensure Benefic II will crit). One I have not seen is "upon crit, get back x% of MP spent".As I think on this a bit more... well, I'm going to be bluntly honest: A lot of healers out there I've seen tunnelvision and kind of suck at multitasking, even just at watching health bars while DPSing. The level of attention that a proc-based rotation requires... well, I think it would make it a lot worse.
It'd take a couple of things. Increase damage taken by everyone or nerf healing slightly, possibly nerf MP regen, limit sources of in-combat rez (this is tied to the MP changes; healers can't blow a big chunk of their MP on rezzing if they can't hardcast rez in combat *taps head*), add aforementioned duties like cleansing and dispelling, and redesign dungeon mobs so that you can teach leveling players how to play within this new paradigm we're introducing.Here's where I agree the most. When I'm playing a healer, I want to, you know, actually heal more.
It's a difficult problem to address not just because of stuff like ilvl bloat, but because of the way battle encounters are designed on a fundamental level.
Sadly I don't have a magic bullet solution. Simply nerfing heal potencies won't do it either - SE seems to have dabbled in that slightly with ShB and well, yeah. It's really hard to think of a way to do this without suggesting a total overhaul of battles, which of course won't happen...
The one word of caution in all of this is that if the goal is to make healing more challenging, there's always the risk of pushing things too far and scaring away the laymen that level healers for the storyline or who dabble but don't main a healer (like me).