> If the anti-missile system is effective, then it allows for a first strike without retaliation. I.e., anti-missile systems are offensive weapons.
This is simply false. An effective missile defense system is not necessarily capable of defeating an entire nuclear arsenal of a peer power - easily verified if the number of interceptors is substantially smaller than that power's inventory. The nuclear defense triad extends beyond just ground launched ballistic missiles - bombers, cruise missiles, and submarine launched missiles all offer means to retaliate which a missile defense system alone can not defend against. And finally, the adversary can also develop an antiballistic missile system for exactly the same protection. Of course in the real world such a perfect missile defense system is not possible - realistically there's always going to be too big a risk of some nukes getting through for any rational actor to seriously consider a nuclear exchange. As for the irrational actor, they can not be trusted to avoid mutually assured destruction.
It's like saying a city wall is an offensive weapon because the one city with a wall can attack others without being attacked. Yeah, maybe the very first city to build a wall had a bit of an advantage before everybody else built a wall, but anyone can have a wall without putting anybody in danger.
> If your adversary is not convinced of this, then they have every incentive to react to (possibly false) alarms sooner.
Without a missile defense system, your only hope of surviving nuclear war is taking out your adversary's retaliatory capability - this means a first strike is primarily going to be tasked with doing just that and thus makes launching quickly in response to an incoming strike a necessity. Even if there is a chance it's a false alarm, a limited retaliation might be seen as a more survivable option than waiting to confirm. If on the other hand your opponent has an effective missile defense system, then they have far less incentive to try and suckerpunch you, and thus an unexpected strike can be much more safely labelled a false alarm. Certainly there is no incentive to retaliate faster - what is that going to do but waste missiles? You don't overwhelm a missile defense system with quick retaliation but with massive retaliation. While nowadays some military commanders might entertain the notion of a limited nuclear conflict where a small strike warrants only a proportional retaliation, with missile defenses it's unambiguous that once nuclear war is triggered people will launch everything they've got in an effort to guarantee at least some warheads make it through, and thus reinforces mutually assured destruction.