I accept because I’d probably forget about them at some point, until I came across the control board whilst dusting or something. That might be a good thing.
However, we need to make sure that “the button” combo is not flippable by random feline actions. Those I can’t control, and they’re fiendishly clever little devils.
I’d threaten people with them but never actually use them . . . or maybe not. Gotta keep them guessing. Maybe I’d just drop conventional weapons on their leaders from my predator drones and see if anyone wants to stand up to me and make me stop.
I can cause significant global financial chaos at work with my keyboard. Adding the football to my armamentarium shouldn’t be a problem.
I need more details. What kind of life do I end up living in this scenario? Can I use my position to extort fabulous wealth and privilege? And if not, can I point out that I now have ALL THE NUKES, and perhaps you should rethink that answer?
We also need to make sure it’s not a “JOLLY, CANDY-LIKE BUTTON!!!” 
Sure I could, you all know what a cautious fellow I am.
casually tosses The Button into his suitcase
I’d set it up on a sidecar attached to my motor-cycle, and wire it to explode if my heart ever stopped.
More seriously, I don’t think I’d be tempted to use it, or even threaten to use it, so in that sense I could be trusted.
If I’m trusted to launch it at somebody if they launch nukes at us, a la MAD, than no, I don’t think I could be trusted.
If you go by all the times I’ve played Sid Meier’s Civilization, the answer is likely a resounding, “no fucking way!”
Especially in Civilization II. Why expend a bunch of units to capture a city? Just nuke it and send in paratroopers. Good times. 
Go through the background check and continuing scrutiny involved with something like the U.S.'s PRP, and also have to follow the ‘two man rule’ all the time? No thanks. And all that for something that AIUI is simply of a list of attack options, black box authenticator protocols, and communications instructions? I like my freedom and anonymity too much.
Christ, imagine if it actually was a system that could remotely send unquestionable orders to start Armageddon. You’d have to live in a sealed bunker buried beneath the seafloor, and that would only be a good start to the security precautions that would be placed on you. Double Plus no thanks.
To be trusted not to use them inappropriately, and to be trusted to use them in a counterstrike.
That’s what’s being asked in the thread title, but the actual poll is just asking “If you were offered power, would you take it”.
The answers to those two questions are probably (hopefully) not the same for the vast majority of people. I don’t actually think I could be trusted with nuclear weapons, but I bet I’d do better than the average person, so I’d accept.
I can’t decide myself if and when to use them? What’s the point, then?
I missed this the first time around, you think the only purpose in having power is to abuse it? To use your memo-board example I could use my power to stick a note on it to advertise an upcoming bake-sale, that is using power but not abusing it.
I’m really not sure I understand you here, if I do then it explains a lot about people and the world.
See, this is the virtue of public polls. I can scan the Yes votes and snicker at the all the people I wouldn’t trust with a pointy stick, let alone H-bombs.
That said, I think I’d do fine. I’m extremely even-tempered, and I don’t hold grudges. The only problem, as someone else alluded to, is that with a thousand incoming warheads there’s a good chance I would just say, “Eh, fuck it; one dead continent is better than two.”
And you’ve already got the passwords…
I could be trusted with them. As long as there is no whiskey. If there is whiskey, one of these days, they’re all going at the same time.