I wonder how confident the US and NATO would be of Russian claims to success in intercepting incoming ballistic targets? Its a good thing that the nukes have been replaced by blast-fragmentation warheads as EMP and radiation hazards are very serious concerns.
The thing is, the US, Israel and China have demonstrated precise hit-to-kill capability. This provides unambiguous proof the warhead has been destroyed, as evidenced by a brilliant flash detectable by space and surface sensors. No wasted shots of precious ABMs "just to make sure". No flash from impact, you know you have more work to do and fire off another interceptor. At least you know the status of the threat.
With high-explosive warheads, what assurance does one have that the hardened warhead has indeed been neutralized. You could keep firing from your limited stock of ABMs until hopefully it disappears from the radar screen but so long as it keeps coming, you have to assume it still a threat right? You'll run out of ABMs pretty quick in such a scenario.
