Each weapon is not a launch site. Most ICBMs have multiple warheads, so taking out 1 site could mean 20 out of commission. Then you have failure rate probably around 20%, them missile guidance errors of a few percent, if they have SDI it can take out up to 70% most likely. Between all that your 1000 suddenly becomes around 5, and those 5 won't be your top targets and some of those 5 may hit the same target multiple times. So by the time you launched all 1000, a lot of your opponent's targets are still there and the beating you just took means you can't build more. During the cold war 1000 only would be idiotic.
What no one is mentioning though is nuclear subs or stealth bombers. Both are able to deliver to targets in non-ICBM ways that are probably much more reliable. With the stealth bomber and subs, 1000 might just be enough for one large conflict.
The other thing not mentioned, a lot of the older ones were called tactical nukes. Something like the Davy Crocket which was launched from a ground artillery piece from a couple of miles away and was small yield. Meant for a Russian tank column. The A-4 and I think F-111 could both carry small nuclear bombs which we no longer have. We don't use those anymore and things like that probably accounted for half of what we used to have.