(this might be a duplicate. Or maybe I'm just thinking of comments to another question)
Because, if they only break the ice that is right next to the island, the water there is shallow, or the distance is easily leapt over.
So, they have to venture out on the ice, a good ways, to "protectively" break the ice. They want to do this before the ice is stable enough to carry the weight of wights, who often have very little muscle, tissue or water weight that normal humans have.
That means, in order to make the ice too unstable for wights, they have to venture out and further weaken the ice before it is stable enough to hold humans, which means they can't do it, physically.
Then there's the matter of breaking the ice. You'll notice that once the ice was breached, the weakness/breaking spread pretty quickly. That means if I venture out and it does somehow support me, breaking the ice out there is very likely to destabilize the ice that is still supporting me, plunging me to my icy death. Even if it doesn't, the randomness and chaotic nature of the ice cracking, breaking and destabilizing means it would be nearly impossible for me to go around and systematically break the ice only where I want it broken, in sequence.