Thing is, the idea of the support role in League is a great one, and it's something a lot of other games don't really realize very well. HotS has a few non-healer supports (mostly specialists like Medivh and Abathur), but CC mage supports like Morgana, Janna, and Rakan(a? :P) are really rare. Vision is also not a thing for most heroes. Zagara has a solid bit but relies on it too much to fight to put it down much outside of lane + teamfights pre-10, Khaza has a little if he goes Air Spirits, Medivh ofc. is half-scout in some comps, but there is no "warding". Blizzard definitely has their own flavor of supporting, and it's almost completely pure healing. Most of the "utility" is knockbacks or things that enhance healing (resurrects, Ana's grenade, shields everywhere, etc). It shows up a lot in OW too. I've not played WoW or Diablo to know how it works there, but I'd suspect it's similar.
I do agree on the income sources thing. The support items were mostly a stopgap to fix that issue, and now the attempt to make them a more effective income source is just taking a bad solution and trying to make it better by doing it more rather than reexamining the system for a better solution. And again, supports who aren't Lulu or Ivern aren't allowed to play anywhere but bot. Remember what happened when people dared try to take Janna mid? :(
I do like the concept of champs designed for duo lane, mind. I mean, look at Rakan, or Nami, or Leona's passive. I think the concept of 5 champs in a 4-location game isn't inherently bad. There just needs to be another viable income source, and we have to be okay with damage supports happening. Take it from adc/support to duo lane, and let supports be as strong as anyone else. The one takeaway from gold items that I really like is the concept of two mutually-exclusive income streams to prevent someone from just taking both and becoming gigantic early.
The idea of quests is actually a really good one, but it's tied up in a bad mechanic at the moment. Don't tie it to an item we have to spend most of our early gold on just so we're allowed to have an income. I could see some sort of "questgiver" NPC at fountain who gave quests where, let's say, you get gold for healing, but lose it for taking CS. I mean, that's a bad example, because that'd encourage people to play stupid just to get some gold, but it makes the point. Heck, you could even go really revolutionary and tie all gold income to quests. Maybe CS doesn't naturally give gold, but there's a quest that gives you gold for killing minions. Maybe someone like Ziggs would rather go for a quest that asks you to damage towers, idk. Basically, give quests that reward players for completing objectives, and make that the income stream. Experience can honestly still be done the way it's done now, with maybe some small changes if we want to encourage more roaming.