I assume that you are referring to channel energy when you say that Pathfinder doubled down on clerics as healbot. Regardless, it's worth mentioning that's only for approximately half of clerics, as the other half do the opposite of healing with the same class features that typically get clerics labeled as healbots.
Magical healing is a necessity in D&D prior to about 4th edition, because mundane healing simply takes too long. It's cool to want clerics to do something other than healbot, but not having healing at all isn't actually an option. This is baked into the design of D&D, and a lot of other games. You can get by with an evil cleric who spontaneously casts inflict spells, because they can always load all cure spells the following day and recover everyone in downtime, but you at least need that option, minimum. It is what it is. I wouldn't personally call this a design flaw in the system itself, but I think that's a fair argument to make.
I'm inclined to be more favorable to your idea*, because SWADE is a lot more generous with mundane healing. Not having magical healing could actually work, because any wild card with a d4 healing has a pretty good shot at recovering a significant chunk of your total wound capacity. You just need a chance to take a breather and pull out the first aid kit. You don't actually need someone who has the healing power. It's handy, but "savvy woodsman with bandages and herbs" is plenty.
It's an interesting idea. I share a lot of your opinions on the matter. I really liked the 2E Forgotten Realms treatment of specialty priests because it mattered what god your cleric worshiped. The gods themselves all have different powers, and that should translate to their priests. The way domains are handled in SWADE does a far better job of making a cleric's god matter than the domains of 3E ever did, especially once you start to work in the concept of trappings.
The only thing in SWADE that I see making clerics a healbot is that they get the healing power plus some amount of other stuff. Giving them three powers rather than healing + 2, or picking some other deity appropriate power for their freebie and forcing them to select healing if they want it is a thing you could do, I suppose. However, if you take away the healing power entirely for certain gods... what makes a cleric different from a sorcerer who worships that god? You could certainly do that, but all clerics have access to cure light wounds in D&D. If you're taking away the healing power from certain clerics based on their god, is this still a SWADE conversion of D&D/PF? The question is a bit philosophical, I'll admit.
*I was initially skeptical, because I've been down this road before numerous times. Someone gets the notion that it would be awesome to have a game with no healing, because healing sucks so lets just not have healers, and 9/10 times the game is dogshit. This is because the person with the big brain idea to get rid of healers didn't actually address anything about the game that made healers necessarily, they just deleted the healing abilities and any classes based around healing. Granted, I see this happening more frequently in video games, but it pops up in tabletop games as well.