I should do an effort post eventually on lasers and laser weapons, but suffice it to say that there are some significant issues with them just on the conceptual level. I will try to list them in order of severity (most significant problem first):

• The critical technologies to run a laser weapon system don't really exist yet. You need something like a 50MW laser (I don't remember where I'm getting that figure from) to reliably blow up a supersonic missile (not even going to get into hypersonic lol). High power industrial lasers are still in the low hundreds of kilowatts, the best the military industrial complex seems to be capable of (according to pop science articles) is promising to get to 500 kW soon.
• Lasers fucking suck to work with and work on, and the parts burn out really quickly, even on 5W lasers. Laser optics have to be kept in semiconductor manufacturing cleanroom conditions, like <1 particle per cubic foot, to prevent burnout, and that's just lasers over 1W. You drop a skin-flake on a mirror while the laser is on and that mirror is smoked out and totally fucked. Ask anyone who does calibration work on lasers about "collimation" and listen to the gripes. It's difficult, precise, tedious work and there's no way an 18 year old soldier is going to be able to fix one in the field under fire.
• Lasers use a lot of power. Like, you see the "50MW" figure? That's just what's coming out of the final lens. It's going to suck down several orders of magnitude more power to get to that figure. "Oh, just turn it on when you need it" I hear you say. Good luck shooting down a missile when it takes the laser like 10 minutes to warm up to operating power. Laser weapons capable of shooting down missiles will be restricted to ships and fixed land-based installations for the first several decades at least.
• Lasers emit light, and that light doesn't fall back to Earth. The weapons will have an "effective range" listed, but they will still cause significant damage along their vector for several times that distance. Any aircraft or satellite in the line of fire will be hit, and, at the very least, the optics on it will be permanently fucked.
• The missiles you're trying to shoot down are moving through the atmosphere. Air does a very good job of wicking away heat, and the primary effect laser weapons have on their targets is creating heat. In fact, it would be easier to shoot down a missile moving laterally (with regards to the laser's point of view) than approaching head-on, as the heat-sync effect is most pronounced at the front of a projectile.
• Lastly, I'll list weather, because it's honestly not as big of an issue as a lot of people make it out to be. Laser weapons will almost certainly operate in either the infrared spectrum or the green spectrum, and in two very specific wavelengths (one green, one IR), as those have the lowest Earth-atmospheric attenuation of any EM radiation, and they would only run into operational problems in severe weather conditions when aircraft and missiles can't fly.