Tony is, to various degrees, an a-hole. That much is a given since the first IM movie. But keep in mind that his character arc in the first film really begins with an epiphany that he should become a better person. If you take historical events that take place before IM then, of course, he's a complete womanizing d-bag, but that's not (entirely) the same "stand-up" Stark we have today. I think of the current Stark as more of a "recovering" a-hole.
Now let's take your issue with Extremis. Your entire thesis is founded on the presumption that Stark is capable of creating a "stable" version of Extremis. That's a huge leap in logic. Being able to understand a process or even "cure" someone from it is a FAR, FAR, FAR cry from being able to modify the process to make it viable in vivo. It's entirely conceivable that it's impossible to use Extremis in the way it's hoped to work.
Now let's take the ethical issue. By "Extremis-enhanced" do you mean Human-torchify the Avengers? Or do you just mean the healing ability? If you mean the former then there is an ethical issue that doesn't apply to super-soldiers. The destructive ability of Extremis is so excessive that its utility is limited. A super-soldier can pound a horde of Nazis or rescue a kitten from a tree. With Extremis you're basically incinerating everything. Put it another way, if you found a cheap way to make nuclear weapons, is it effective to get rid of the soldier's trusty M16A4 and equip each soldier with a nuclear weapon? I don't think so. Extremis is impractical.
If you mean to just to grant the healing properties of Extremis to the Avengers then it's still a problem. How do we know it wouldn't have long term adverse side effects in the future? In 5-20 years it may cause anything from leukemia to premature balding. Is that different from Steve Rogers in WW2? Yes it is. Remember, the super-soldier was an experiment in a time of war. As a subject, this scrawny guy from Brooklyn was expendable. The Black Widow and Hawkeye are not "expendable" (I know what you're thinking - don't even go there. They are active members of the Avengers so, by definition, they have utility).
So yes, even if Stark has a theory of making Extremis a viable healing tool, it would be unethical of him to test it out on his buddy. Remember that Stark is a tech-guy - he's not a scientist, like Banner. I have more faith in his ability to make mechanical prostheses than medical interventions that bypass the entire FDA.
Pepper would probably argue how much he's "reformed" given they're on the break. More presumption I guess.
Aside from Killian, did any of the other Extremis soldiers blow flames out of their mouths? Yes, they got pretty hot... and used that heat to their advantage in battle. I don't see how it's excessive... until they blow up. In the final battle of IM3 none of those soldiers blew up the way the guy did at the Chinese Theater because he didn't have his dose of the "regulating" drug. They're not incinerating everything... otherwise they wouldn't be able to wear clothes. His crony wasn't melting the chair he was sitting in when his boss was having a meeting with Pepper. If they're melting things, it's intentional. I see a lot of utility in that. The Fantastic 4 would probably agree it's not to have that ability on your side.
Yes, they did the Super-Soldier thing during war. A war between people. Now we have aliens invading the planet. It might not be an "active" war, but it still requires trying new things before the next major conflict (Infinity War?). Could it go wrong? Sure... look at the super-helicarriers in Winter Soldier. Could have been a very valuable asset... if they weren't going to be used to annihilate the population of half the planet. So yeah, I would have concerns about who is going to be Extremis-ized. I certainly wouldn't want Ross heading up that program.
Does Extremis have negative side effects? I know growing limbs back is a pretty sweet side effect that I'd be at the front of the line for. If there's unseen negatives I can only assume Stark is still conducting lots of tests on Pepper to keep an eye on that, right? She had it. He got it out of her. If she's not going to be in the movies any more, it would be interesting if they killed her character off by saying there were unseen side effects to having been "exposed" to Extremis. But they haven't. So she's had a clean bill of health for at least 3 years. So, again, why wouldn't be try to use that science to heal his buddy?
I agree Hawkeye and Widow might not accept the offerand I considered that before, and for the same reasons. Clint's a family guy and semi-retired. Natasha was already messed with a lot as a child. At a minimum, though, Rhodey probably would. Given his current condition, he'd be a fool not to. Again, Tony understands the science of it and obviously knew enough to fix his girlfriend. Add to that, he has other brains to help him, including the artificial lifeform he was responsible for creating with Vision. Seems like a waste.
I'd argue how expendable their team thinks they (Clint/Natasha) are, given they send unenhanced people into battles against hordes of aliens and an army of robots... wearing no body armor and using a bow and arrow and handguns. They trust that they can handle themselves, but, like you said, Stark doesn't (that we know of) offer to put them in any kind of Stark-tech. How ethical is it to allow people to go into combat equipped disproportionately with their enemies? What would the world think if we started sending troops into Iraq with no body armor and only handguns and longbows?
