It's both.
The pickups have more to do with the frequency response of a guitar (though the guitar will have a large effect as well), but the guitar's design and the woods used have a larger effect on the guitars "envelope," which is to say, the shape of what synthisists would call the ADSR (Attack-Decay-Sustain-Release). Bolt-on necks will tend to seem to have a stronger attack (which is actually a function of the quick decay on a bolt-on neck - kind of like using a compressor to increases the attack on a snare drum by using a slow attack), while the attack on a set neck will seem weaker, with a neck through seeming even weaker (because there is a slower decay). But of course, the sustain on a neck through is much longer on a bolt on. And of course, the pickup also has some effect on the envelope as well.
But yeah, just basically, it is the whole package.
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