The only class that really and truly fits this role is one of the ones you dont want. BM hunters have a pet to tank. They have 1 button AoE. They have short CDs, so messing one up is forgiving. They do solid Single Target for Raiding, and consistent sustained DPS for M+. With Ferocity pet, plus mend pet rolling, they do a lot of passive healing. AND brings Lust.
The only other spec which may fit, would be destro lock. But Destro locks, rely a lot around their infernal CD. Which means, a poorly timed infernal, kind of bricks you till its up again.
Ele shaman wont fit because of the opposite reasons. It has short elemental CDs, which makes it more forgiving, but it is fast paced, with a lot of buttons going on. Even more if you play icefury, which I dont enjoy. But it brings Lust.
Frost mage would probobly fit what you are looking for. It is fairly straight forward. It is less fast paced than ele, and more forgiving than destro. But no pet, so you have to kite. Which you will be amazing at as frost. The only problem is right now frost is the least desired of the 3 mage specs. But it brings lust, which is nice.
You could also look at Boomkin. While it isnt the bell of the ball right now, it can still put up decent numbers, and is fairly straight forward as a spec. There are some easy talent choices, and it is pretty forgiving if you play nightfae. If you want to play venthyr, then you are purely a CD class, which I would not recommend. With resto affinity, you can do some good healing, and as a class, you are fairly tanky. You have root and treants to help solo elites.
All things considered, even though you say no⊠BM hunter is really what you are looking for. And if you want to juice up the playstyle a bit, play venthyr. Its not Meta for the spec, but it makes the rotation a little more fun and interesting.