Too many people on this thread are confusing 'feeling safe' in our real world political sphere versus what is actually present in the MCU-world political sphere, which, I gotta tell you, the MCU-world is one messed up place to live. In the MCU-world political sphere you'd almost HAVE to take Steve's stance to survive because that place is one district short of the Hunger Games.
Iron Man 1 - weapons of mass destruction being sold and bought in the black market.
Incredible Hulk - Ross was running a GOVERNMENT FUNDED human experimentation program. (side note: Ross was responsible for a whole lot of collateral damage in both Brazil and Harlem in that movie, which made his speech to the Avengers in Civil War remarkably hypocritical).
Iron Man 2 - the US government wants the Iron Man suit for their own purposes.
Captain America First Avenger - Steve 'dies' with the tesseract, which not only has the reality gem buried within it but that humanity knows nothing about but feels entitled to use anyway, it is uncovered and, fast forward to Avengers, used by SHIELD to make advanced weaponry, which, according to Thor, let's the universe know that Earth 'is ready for a higher form of war'. Red Skull starts the evil, international Hydra organization, which does not die with either him or Steve and, in fact, continues to grow and prosper.
Avengers - this "World Security Council", whatever that is, decides the best way to handle invading aliens is to drop a NUCLEAR WEAPON on New York, population 8.4 million.
Iron Man 3 - the Vice President hatched a plot to kill the President. Extremist, an experimental procedure, was being tested on soldiers. 'Big pharma' had their hands all over the highest government office of the United States and were, actually, controlling things.
Captain America Winter Soldier - Not only did we find out SHIELD had been infiltrated and run by Hydra since its inception, but we also found out that there were US senators who were Hydra AND that Pierce, Mr. Hydra himself, was once a world secretary to the aforementioned World Security Council and had been offered a Nobel Peace Prize at one point in his career. Zola also hinted that Hydra had been behind many wars and 'changing history', which indicates Hydra's influence was not contained to the United States.
Civil War - someone thought it was a good idea to promote "I-created-Abomination" Ross to Secretary of State. And since that position is filled at the discretion of the President I'm going to assume that President Ellis, in the MCU, is as screwed up as his Vice President and Senators were. Not only that but the Raft was fully built, and fully functional, which meant its construction had to have happened before the Accords were a gleam in anyone's eye... just things that make you hmmmm.
And this isn't even covering what's going on in the Netflix shows or Agents of SHIELD, which is even more horrifying. In Jessica Jones another 'Big Pharma' organization is experimenting on humans, which results in Killgrave, Jessica, herself, and Nuke. In Luke Cage arms dealers where using Chitauri debris to make horrifying weaponry, specifically targeting supes, who are now, thanks to the accords, not allowed to defend themselves. And in Agents of SHIELD Inhumans are required to register which puts a huge massive target on their backs as they are actively being hunted.
So yeah, honestly, I think Steve's stance was entirely warranted considering all of the above. You stand by your river of truth, Steve, I've got your back there, buddy.