There's been a couple of concepts that tried making their "autoattack" a free heal on allies or one that did no actual damage, but instead started stacking a slow effect on someone that could get up to like 75% going 5% at a time. All these ideas were generally frowned on for being simply infuriating to play against.
Stacona is the one I remember that made a healing autoattack. What I was trying to explain to him, though I probably came off as crude, is that healing someone "on hit" for free can get really infuriating if someone builds, say, nashor's tooth. Nashor's, would mean spam-healing a single target through a pretty crazy amount of damage for free.
That's not even including the rest of that champion's kit, which included another huge heal (that fortunately did have a mana cost) and a substantial shield effect. A champion like that could literally make anyone an immortal being.
The other thing about champs with no true autos is that they're actually pretty unfun. All the ideas I've seen essentially are supports who have literally no reason to step anywhere near an enemy champion, and instead just sit back and spam heal their allies. I call it "old Soraka syndrome" since before she was reworked, she literally didn't have to aim a single thing: her Q just hit anything near her, her W and E were point and click hp and mp returns to her allies, and her ult was...well, that didn't change. Needless to say, old Raka really didn't have to do much of anything in game besides spam her laner and let them do all the work.
The exception was the one that slowed on his autos without dealing damage - that one was a very in-your-face concept that I would definitely have loved to see had their abilities not have been totally god-modded...but that one is buried somewhere in the archives, and I'm not digging through that. xD