Approaches modifying skills
You could leverage the mechanism of skills modifying other skills but, in this particular case, approaches modifying skills. Each character would have their set of skills plus the basic six approaches. Whenever a skill is used the one who uses it has also to select one of their approaches which modifies their roll.
When Sneakily striking an opponent who does not expect an attack, roll fudge and add Brawl skill modified by Sneaky approach. When doing a strike and trying to impress crows at the same time, use Flashy approach, and so on.
This will most likely noticeably slow down the gameplay but will increase the granularity and simulationism of your campaign.
Tag with Aspects
Remember NPCs can create advantages, compel aspects, etc. If someone's acting in a very odd way it's likely that other characters can start tagging him with aspects disadvantageous to him. Someone that's doing everything flashy could easily get a "Showing off" situational aspect created on them that makes it easier for people acting sneakily or carefully to beat them in a contest (user can invoke advantage for +2)
That said: Change the Contests Away from their Advantage
When someone defends, the ways they can reasonably go about defending depends on how you attack. Someone has a harder time defending carefully if you're attacking quickly or sneakily, since it takes time to be careful. Similarly, avoid attacking a flashy opponent forcefully or sneakily, since these are the more reasonable times when they could use their preferred approach. Instead: be quick, be clever (the sample text for clever even says "finding a weakness in the opponents swordsmanship")
As a GM, you have every aspect and approach arguing power the player has, and an easier time approving your reasoning since it falls to the GM to approve approaches.
Of course this part relies on you not getting taken out before you have a chance to act, but hopefully your major villain can't be gibbed in a single action, give them stunts to act first or otherwise improve their survivability if you keep getting taken out before you get a chance to act.
Ultimately : Loose rules systems thrive or fail on the resolve of the GM
Basically, FAE's rules can be pushed any which way by someone that talks cleverly or imploringly enough, as the GM though you get to say though whether some reasoning is good or not. Same thing when it comes to aspects and stunts. As you noted, the game can't withstand it very well if someone can use +3 on everything. And if the other players aren't playing that way, it diminishes its value to the game. So you have to say no some times.
From the book :
So your first instinct is probably to pick the action that gives you the
greatest bonus, right? But it doesn’t work like that. You have to base your
choice of approach on the description of your action, and you can’t describe
an action that doesn’t make any sense.
Roleplaying is about keeping things fun and interesting. I've seen sessions be interesting and enjoyable laughfests because everyone was on-board with figuring out how to best solve problems according to their character's styles. If fun is being had, let them be a little crazy (and join in on the crazy with your NPCs), if things aren't being fun, it's time to put the foot down.
Best Answer
When it comes to mechanics & narrative, Fate can be a juggling act
For as much as narrative descriptions are the lifeblood of a game of Fate, you do need to think about the mechanics, too, to get the game to run smoothly. It sounds a little like you are focusing on some of the mechanics, but in the wrong spots. I think you need to switch your focus.
Narrative descriptions such as in @Christopher's answer above, are great examples of ways to mix up the narrative portion of "avoiding a laser", but you and your group can easily slip into a rut of just shoe-horning "I highest-approachly dodge the laser" as every defense, which may not end up being very satisfying.
Step out of the act and into the intent
You need to separate the why of an attack from the how of it being carried out.
Fate can be a little different from other games in terms of how combat works. It can be an easy habit to treat it like other games which model a specific task at a specific moment i.e. a laser is coming at me, what do I do?
As much as attacks in Fate represent this action-to-cause-harm, what they mechanically are is a little different. An attack in Fate is an attempt to move an opponent closer to Taken Out. Period.
This frees up your reactions to focus on "these cyber-wolves are trying to kill us with lasers!" and not "I have a specific laser coming at me right now".
Put it back together
There's definitely a difference between "How can I avoid this specific attack?" vs "How can I make an attack against me fail?" The former is reactive, while the latter is proactive.
This paradigm shift should help you understand how to bring some of the other approaches into the narrative. Approaches that seemed implausible as answers to the first question could still be good answers to the second question.
You may be hard-pressed to describe carefully dodging an attack, but you can survive an attack by carefully avoiding it with good use of cover.
I've never heard of sneaking out of the way of an oncoming laser, but I can understand how you might sneakily trick a cyber-wolf into shooting lasers where you're not standing.
For example:
Compare
GM: "The wolves shoot a laser right at you!"
Dice happen, the wolves fail.
PC: "I cleverly see it coming and get out of the way!"
with
GM: "The wolves bring their lasers to bear at you!"
Dice happen, the wolves fail.
PC: "I've noticed the lasers come in 2-second intervals (clever) and time my movements between those intervals"
Additional examples
GM: "The wolves bring their lasers to bear at you!"
Dice happen, the wolves fail.
PC: "As the wolf's tail comes up, I shoot out the closest wall-outlet. It spews forth a shower of sparks (flashy), confounding the wolf's targeting computer"
GM: "The wolves bring their lasers to bear at you!"
Dice happen, the wolves fail.
PC: "I toss my scarf out from the right side of the pillar while I dive to the left (sneaky)"