To OP mostly: I think that you more or less hit the right target with what is wrong with clerics. They are too focused on one type of heal. The magic missile = cure light wounds is a little off but fairly fitting.
I disagree with your solutions and premises beyond that however.
The solution presented turns a cleric into something else, sustained buffs that they have to concentrate on seems to be problematic, as [MENTION=42582]pemerton[/MENTION] says. It also strikes of being too dissimilar from what clerics, imho, should be doing. They shouldn't be sustained effects, like a bard's singing might be, they should stick to healing(or cursing) and then extra powers based on deity.
I think that part of the problem is that many people at WotC have become too focused on what cleric is at its core without looking at further options, or even options that have dropped by the wayside in previous editions and bringing those back.
What I would personally do is something along the following; make them more into a white-mage of my old gaming roots. Give them healing, make it class related not necessarily spell related. This may mean they have spells and healing and "turn undead" abilities listed. Then, give them spells but the spells should be restricted or enhanced by the deity they have. A fire god grants his clerics fire spells, for combat or other uses similar. The healing (or cursing) is completely secondary, but could be in basic form for backup healing. Then if a cleric wants to get spells that heal give them extra effects.
This would be like the wizard who gets a basic spell at-will that damages an enemy. Give the cleric a basic at-will ability that heals people.
But like I said, and you alluded to, make the healing spells different. Not all spells need to be the same basic spell with more power (cure light, medium, serious). Give them different forms of healing, ones with extreme range, ones with extra effects, ones that combine healing with regeneration or a buff of some kind. Let them combine different kinds of healing at higher levels, combine a stronger form of healing with a longer range or bypassing XYZ or always hitting or whatever.
Do this on top of deity related spells.
Then compare that to wizards. If wizards get scorching ray, flaming sphere, wall of fire, meteor swarm, fireball, etc. Then clerics should have as many varied options that aren't just increasing the amount of healing dice that spell provides.
Just my two cents.