Another thing is that the system doesn’t shoot down every incoming rocket. It predicts where the rockets will fall and doesn’t intercept them if they won’t hit anything important. The incoming rockets are unguided, so it’s relatively straightforward to figure out where they’ll land.
Indeed, there’s a decent short video of the systems involved in Iron Dome, how it works, and how it decides what to intercept and what to ignore here:
The basic upshot is it will only intercept projectiles headed for populated area and will attempt to intercept them such that there is the least amount of debris landing on populated areas from successful intercepts.
You get better results if you use your explosive warhead to launch heavy chunks of metal at the target. The blast wave will attenuate over a fairly modest distance, but the chunks of metal will be able to cause damage at a substantially greater range (or over the same range for the cost of less explosive, which can mean a cheaper and/or more maneuverable missile).
Can’t vouch for the veracity of this, but here’s a Reddit thread showing what is claimed to be a Patriot missile warhead, with metal cubelets packed around an explosive core:
https://www.reddit.com/r/MilitaryPorn/comments/emfc4s/patriot_missile_warhead_785x679/
Another warhead variant here. Different implementation, but it’s still about using high explosive to launch metal toward a target:
The Tamir reportedly does something similar, in that the explosive is designed to propel the fragments primarily in a radial direction rather than spherical (as the Patriots do).