How would you handle it?
The Amp Specialist aesthetically and from a mechanical standpoint seems like he'd be a glass cannon, since they're bound to a wheelchair and barely responsive in any way other than violent space magic (though maybe they do charity work building houses in their spare time, who knows). Minmaxed into oblivion for offense, in fact. So probably my answer would be to let them keep the shitload of will points they probably have, but limit their multitasking skills severely so that they can do only one thing at a time, or maybe just limit the defensive potential of amp use. Make it so they can't float around without being defenseless at that moment, or set one guy's guts on fire while screwing the head off another one, or maybe just make it so defending yourself with an amp is a very impractical thing to do.
Also, like I said, amp-sight, without knowing the Big Ugly Terrible Truth, sounds like a cheap asspull to me, as it is an unexplained method through which Amp Specialists circumvent the human limitations of the amp, in that you need sensory ID of a target or at least an idea of their location to target them in the first place. So I wouldn't keep that as a thing that's possible, given how cheap it sounds from a gameplay perspective, and keep just their limited human senses as their method of orientation. And maybe even not limited human senses, but give them cameyes or sonars or something of that nature, depending on their readiness when deployed.
These limitations - frailty and inability to multitask (or at least limited defensive abilities) - leave the Amp Specialist still very powerful, since space magic is still extremely powerful, and he's got a practically unlimited supply of it, but also probably dependent on support from their squad, since while burninating hostiles they can't protect themselves and vice versa, and their bodies are atrophied enough that a stiff wind could snap them in half. It makes them actual combat
specialists, while the Amp Specialist we saw on Hephaestus is an island, tactically speaking, with no need for support of any kind, at least not for the immediate engagement they're in, and that makes them perilously boring and annoying to deal with.
And if you feel that would make them too easy to deal with, well, that brings me to the next idea, which is the abandonment of cinematic tactics (swinging heat-blades, throwing shit at people, other visually impressive shenanigans) and going for the best effort-to-result ratios in attacks (like boiling somebody's medulla oblongata or raising their heart's temperature by some sixty kelvins). It seems kind of weird that people who've been trained (I presume) in the use of amps by professionals would operate in that manner, at least to me. I mean, pretty much every piece of advice given by the AM in terms of amp use has been categorically against this kind of nonsense. It makes sense for heatblades and stuff like that to appear in convict-performed overshots, since they're not expected to know any better and their enthusiasm often makes them more likely to use maneuvers inefficiently, but I can't see that same sort of naive enthusiasm in a guy who's almost literally been honed into a living weapon by a vicious totalitarian regime.
Then again, maybe they just really like their jobs a lot and have a flair for the dramatic, and are really a good approximation of Stacy put through military training. I don't know.
As for demigod friendly NPCs, I probably would scale back their demigodly natures. Not make them absurdly powerful, but perhaps reasonably powerful. The AM doesn't need to be an inhuman amp demigoddess with 30-something missions under her belt, stunning good looks and a seductive aura of deadly competence to keep players in line, maybe just somebody good enough to have a matter conversion amp and a mass manipulation amp, plus a propensity toward snark, a bit of common sense and a tendency to not suffer fools gladly. The Doctor, similarly, doesn't have to be a horrible eldritch being, since he'd quite frankly be way more intimidating if he was just a guy who's really good at what he does this entire time. I don't think it'd make them less interesting as characters, though that's really how I would handle it, and to be fair, for the Doctor the whole eldritch being thing seems way more relevant to the role he fills than the amp demigoddess thing seems for the AM, though once again that could just be the way we haven't found out how the AM fits into the bigger picture.
Of course, I'm just working off what I know. ER's universe, history and the core concept of space magic are still largely a mystery to all but a couple people who you've told the Big Ugly Terrible Truth to (at least that's what I assume you've told them to some extent). Maybe it all makes sense, as I've mentioned, and I'd probably believe you if you said it did if the whole truth is known.
How do we reign in the power of the high end character?
Or, let me put it to you like this: is it the problem that characters are superpowered, or is it that only certain ones are and they always seem to be the most useful. Ie Harry, did you dislike the Amp Specialist inherently, as an idea, or because it was an enemy that ordinary soldiers and players couldn't fight, one that basically had to be combated by Jim and Miya and their ilk?
Because if it's the first, you're kinda out of luck. The tech is there and both sides are gonna keep using and improving it, especially now. By the time the war ends, in whatever way it does, we're gonna be using artifact level equipment and battles are going to seem very odd in hindsight.
But if it's the second, then that might be exactly what you want, since it will mean that soldiers and average people will have a much better chance of being able to take out major threats.
It's not a problem at all that player characters are superpowered - they clambered through the ranks and earned their tokens fair and square - or that some characters are very powerful (the Doctor I'd hold up as a positive example). The Amp Specialist I dislike because there's an unsubtle undertone of cheapness to his abilities, he's beefed up beyond any form of reason without any form of (non-fluff, I know he probably doesn't like being filled with IVs and catheters, or looks forward to them being reinserted when he hovers out of his wheelchair and spreads his arms in a dramatic gesture, unless he's actually Stacy as previously mentioned and can't help being utterly chipper on the inside) drawback for it. Same for the AM. The problem is that there's no real rhyme or reason to fighting the Amp Specialist, there's just leveling enough firepower his way (and unlocking secret forbidden techniques through the power of seeing someone you sorta like die) until it feels like he's had enough (and the moment is suitably dramatic) and we've all learned a lesson today and also filled up at least three volumes with the big fight. Am I making sense here?
On another note, I'm not quite sure how people are taking away from this that high-powered player characters are somehow wrong and awful. There's, like, five of them who actually do much of anything anymore, and I'm not sure Miyamoto properly counts, since he's very courteously taken to administrative work, only two synthflesh people are actually doing balance-relevant stuff for any length of time (Auron and Pancaek, and both of them are fairly humble in their abilities), Xan's crippled until he kickstarts himself a decompensator and people with MK3s are only really distinguished by the fact that they can fly. I was just saying I really find it funny and inspiring when high-power player characters get mutilated, mess things up or ideally die horribly, especially if less powerful people are nearby and they immediately flip out, and came up with a further rationale extrapolated from Sean's interesting guess on the viewpoint. And then my seething hatred of the Amp Specialist and all he represents took hold of my fragile mind.