Why bother?
Well, first of all, it does cover a portion of the moon (though probably not close to the whole thing). There’s some suggestion in (now not particularly canon) novelizations that it might encompass the whole moon, but the film image does look pretty clear.
More importantly, though, the Empire is not known for their concern for collateral damage. As long as the Death Star is shielded, and as long as their shield generator is protected (which it is; it’s covered by its own shields), why do they care what damage is sustained by the moon of Endor? The Emperor basically wants the moon to get damaged:
The shield generator could have been built in any of thousands of
desolate, lifeless planetary systems. But the Emperor himself picked
this spot from several suggested by Imperial engineers.
The engineers liked the idea of burning up the moon’s resources to
fuel the shield.
And the Emperor liked the idea of crushing something beautiful.
Beware the Power of the Dark Side!
Presumably the shield generator over Scarif is more resource-intensive, but necessary because the whole planet must be protected. However, when it comes to some useless forest moon? Who cares.
Not to mention, the whole thing is a setup:
All the secrecy, all the comlink jamming, all the biker scout
chasing—it’s all been pointless. The whole thing is pointless.
When Han Solo and his strike team reach the shield generator, they
will find a large force of troops waiting for them. They’ll be
captured or killed and the shield will remain in place. The rebel
fleet’s attack—and the Rebellion itself—will be equally doomed.
Beware the Power of the Dark Side!
If the goal is to set a trap for our intrepid heroes, Palpatine doesn’t want to make it too hard.