But during the elevator fight, he's not trying to kill anyone and probably really doesn't WANT to kill the guys attacking him. He takes more strikes to defeat each individual and struggles more with them, indicating that he's pulling his punches so he doesn't just turn their brains to paste like he could.
True and soemthing I hadn't actually noticed; Cap is more than capable of kicking those guys right through the evlevator glass but he did not.
At that point he didn't know that they were Hydra agents. He knew there was something wrong with SHIELD but he didn't know how bad it was or how deep it went. For all he knew the STRIKE team might have been just working under orders and were otherwise decent SHIELD agents.
That's exactly it, in fact. The HYDRA corruption was 100% unknown to him at the time, and all he knew is that there was at least one mole in the agency. He was even buddies with Rumlow to some extent, at least in the sense of professional respect. Once he became aware that the guys he was fighting were actually bad guys (and not just misaligned good guys), you'll notice that he immediately stops pulling his punches and goes right back to killing.
I think the MCU Cap is actually a superior "boy scout" to many of the traditional depictions of Big Good superheroes, where they refuse point blank to kill. Steve acknowledges that it's not possible to non-lethally deal with every single obstacle or bad guy, and sometimes you have to take a life, directly or indirectly, to save innocent and good people. He won't have as much of a problem with the same bad guy getting Joker Immunity and coming back over and over to haunt him unless the decision is taken out of his hands. His Big Good reputation comes through in his idealism and desire to legitimately help people without harming other innocents in the process, and his refusal to let even the good guys rule through fear.
That's one of the things that annoyed me the most with Man of Steel: Superman tries so hard, at least on paper, not to kill anyone and has a meltdown when he has to kill Zod to save civilians. The thing is, he willingly engaged in combat that resulted in obscene amounts of destruction that almost definitely resulted in the deaths and maiming of scores of innocent people. He directly caused almost as much destruction as the bad guys and his high-speed fighting around Smallville and Metropolis likely had lots of casualties behind the scenes: shrapnel from explosions, high-speed debris from a Kryptonian getting thrown into a building, structures collapsing, etc. Then there's his destruction of livelihoods with his tendency to use anything in the environment as a weapon, wrecking homes and businesses and destroying property. But he shows zero care and never once acknowledges the sheer amount of devastation that he has personally caused, to the point of making out with Lois Lane in the middle of what resembles a nuclear apocalypse before realizing that Zod was still around. He only starts to care when he has to personally take someone's neck in his hands and break it.
In short, Man of Steel's Superman likely has some major mental disturbances even before the fighting started.