Missiles these days all have solid rocket motors. The motor continues to burn until it has exhausted its fuel or if it hits something then the fuel will explode or burn. If the missile has a proximity fuse and the warhead explodes, the solid fuel still burns or explodes as well. The solid motor explodes because the collision or warhead detonation will damage the solid motor so that the burn rate increases thus the explosion. Even after previewing this it sounds confusing but I'm tired and too lazy to improve the writing.
Whether the missile has a solid fuel motor or not isn’t the point. In many cases, the missile has a boost motor to get it off the launcher and rapidly accelerate the missile. This motor will burn out and a sustain motor will continue to power the missile. Eventually, this motor will also burn out. The missile will continue to fly and can be further guided to the target.
The point is this. If the missile sustain motor burns out significantly below 13,000 feet, then no one saw a streak fly all the way up to the 747.
Any incremental increase in lethality due to unburned propellant is not likely to be significant. I don’t know of any air defense system that has a design requirement for unburned propellant at intercept.