I think you are probably correct about the targeting of ICBMs, but not sure that warhead yield is very adjustable.
That said, I think the technology for having a rapid launch capability for an asteroid intercept mission is entirely feasible to establish with today's technology. We just need to put the hardware together and test it, so that, like the ICBMs, we can launch quickly when a threat is detected.
The warhead, or impactor, or whatever deflection method will be employed, is another matter. We really don't know enough about asteroids, and the variability of the asteroids, to know what we want to do to change the orbital trajectorie of any particular one. And, then there are comets, too. We may need a variety of things for the payload of our asteroid/comet defense system missiles.
The DART mission was calculated to alter the target's velocity by a certain amount assuming that the impactor just had its momentum absorbed by the target. But, it looks like what happened is that the kinetic energy of the impactor created a crater and threw the material that had been in it directly back along the impactor's trajectory, creating a thrust that also slowed the target's velocity. And, that thrust momentum effect from the ejected asteroid material was more than twice the effect of the impact momentum. So, perhaps an explosion of the right magnitude couid be more effective than just hitting the target with a heavy object. But, we need to be careful to not blow the target apart and create a swarm of pieces that could still hit Earth as large enough chunks to cause extreme damage.
Knowing what level of explosion will work best will probably require us to learn a lot more about both comets and asteroids, so that we have some idea what the target will be like if we find one headed for us that we failed to detect years ahead of impact, so that we could study it in detail before deflecting it. That secenario is always going to be a possiblity, because things in the Kuiper Belt, Oort Cloud, or interstellar space can potentially enter the inner solar system at any time with no advanced warning.