The faster a missile travels through the air the greater the pressure upon the missile by the atmosphere. If it's doing Mach 15 then slight damage to it could destroy it. The ABL can punch a basketball sized hole in a regular missile. I doubt this system could survive that.
Those warheads are pretty hardened. They certainly couldn't take a direct hit by a kinetic kill vehicle but might or might not survive a near miss by a conventional ABM. The ABL has stabilization issues that are trying to be overcome. The main worry about lasers is that they could polish the surface to a mirror-like gloss to reflect the laser-shine or they could rotate it. To be honest I don't know if they could actually rotate a missile effectively enough to reduce laser-shine or how powerful the laser would need to be to oversome such countermeasures. But frankly that stuff is a long way off if ever (Is anyone worried if the N.Koreans can rotate their mirrored missiles?). The fact is a deployment of off the shelf SBL technology right now would be a strong deterrent.