In your scenario (the one you were recommending the ICBM for, to be exact), you are talking about a limited strike on the contiguous United States, using an ICBM with a conventional warhead.
The United States of America has a Launch On Warning nuclear capability. Their current communicated doctrine says they
don't have a LOW
posture, but there is enough ambiguity left in. On purpose.
This means, if you launch an ICBM @ the US, you are getting nuked in retaliation, most probably before the missile even lands. They have no way of recognizing what type of warhead your missile uses.
Historically speaking, nuclear countries have thought about this in the past. My reference to PGS was exactly that, as an example.
Hope that helps.