The best way to make healers fun to play is to make their profession matter in the culture of the setting and in the conflicts the group faces.
Is the character merely "Joe, with a Great-level Healing skill and a Good Herbalism skill"? If yes, why? Wouldn't it be more interesting to have the character be "Joe, an Adept of the Scarlet Order", with contacts in every city, a thorny relationship with ignorant village 'healers', and a respected profession that opens doors in polite society in ways a sinister sorcerer can't?
Make healers be more than just a skill. Tie them into the setting in ways that are distinct from the ways magical healers fit into the setting's cultures and society, and they'll be fun to play because they'll have opportunities unique to their status and reputation.
Combine approaches and extras to create different narrative justifications for powered and unpowered actions.
Because of the limitations of approaches this probably won't be really viable for a long-form campaign story lasting months-worth of sessions. Still, if you're going to be playing shorter games (a month per campaign, tops) this is an elegant option. If you want to use skills, read this anyway because I'm going to bring skills into it at the end.
I've experimented with this for a werewolf game, actually: by using approaches, your characters can have the same problem-solving attitudes regardless of their form: a Forceful cowboy turns into a Forceful werewolf, and a Clever schoolgirl becomes a Clever magical warrior. When they change form, the narrative shifts to give them justification for using their approaches in more magical and combat-oriented ways: a cowboy can't bite people but a werewolf can, and a schoolgirl can't create magical illusions but a magical warrior can.
(This doesn't work with skills because skills represent what-things-you-can-do-ness while approaches represent how-you-do-things-ness.)
You can represent this shift narrative by clever use of aspects and extras. Aspects should generally be phrased so they're relevant in both forms, which can be difficult but gets easier with practice. It's nigh impossible to give generic advice for specific aspect creation needs like this; the best I can say is that focusing on personality and relationships makes it easier to keep aspects relevant between forms.
Now, extras! There are many ways to craft extras around this. (Extras are for when you want to give effects that stunts can't handle, either because it's too many stunt's-worth of effects, or because a single effect is too complicated or powerful for a stunt to handle gracefully.)
Extra: Moon Infusion
Permission: An aspect indicating your magical nature.
Cost: At the start of each session I'm in, the GM's pool of NPC Fate points increases by 1 for each action I can use magically.
Benefit: Because I am secretly a champion of the Moon goddess, once per session I can reveal my warrior form. When I do, I gain the aspect Infused with the Light of the Moon (with one free invoke) for the rest of the scene.
This gives me magical context for using my actions (like flying, and shooting rays of cleansing light). When you take this extra, pick the actions (Overcome, Create Advantage, Attack, Defend) I can use magically.
I chose to make the extra's cost a "make the NPCs stronger" effect (increasing NPC Fate points) rather than a "make the PC weaker" effect (reducing PC Refresh) because, frankly, it's more interesting to face stronger opponents than to have your own power balanced out. In play it's effectively similar: the opposition scales according to the power the PCs bring to the scenario. The exact cost may need a little tweaking depending on your game. I've borrowed the basic concept from the atomic-robo RPG, which is a wealth of resources for this sort of thing.
Don't worry about including "running out of power" type mechanics in these extras: that's what consequences are for. Just as a gunfighter might take a mild consequence of All out of ammo, a magical girl might take Overcome by doubt or Cut off from my power.
But what about skills?
I started with approaches instead of skills because they're easier and more obvious to use with transformations.
On the face of things it's still pretty straightforward: Just as having a loaded gun lets you use the Shoot skill, having sparklemagic attacks lets you use it too. However, a schoolgirl probably doesn't have a lot of ranks in Shoot, so we need a new level of complexity in representing the transformation if your game uses skills instead of approaches.
Extra: Moon Infusion
Permission: An aspect indicating your magical nature.
Cost: At the start of each session I'm in, the GM's pool of NPC Fate points increases by 2.
Benefit: Because I am secretly a champion of the Moon goddess, once per session I can reveal my warrior form. When I do, I gain the aspect Infused with the Light of the Moon (with one free invoke) for the rest of the scene. This gives me magical context for using my actions (like flying, and shooting rays of cleansing light).
My abilities are different when I'm a warrior: when you first choose this extra, shuffle my skill ranks into a different configuration representing the talents of my magical form (the new configuration must still follow all the game's rules about skill ranks and caps). Whenever I reveal my warrior form, my skills change to their new configuration. They return to normal when I do.
Now we've got a girl whose abilities radically change but her aspects stay the same--so she's still the same person, but she has a different set of competencies when she's transformed.
Best Answer
A few major points:
As for the commonality of magical opponents, it really depends on what level and size your party is. As a DM of (now) level 5 characters, we are barely fighting any 'magical' opponents. but at higher levels, I believe that this would change.
Resistance is HUGE. Especially if your character is a high AC character, disadvantage is a massive benefit. And if you have resistance to all non-magical weapon damage (Which is most of the damage you will be taking), you will be basically indestructible.
As for your DM finding magical opponents so as to balance this, this would just put unnecessary stress on your DM, and greatly limit the variety of opponents that he could use. As well as this, wouldn't it just be better to not have the resistance and be faced against a greater variety of opponents?
TL;DR - Magical mobs and enemies are in general, very rare, and difficult for the DM to use. Having resistance to non-magical damage would be supremely overpowered and in general just a bad idea.