This might be getting pedantic, but pickups can't have "loose" or "soft" or "tight" low end. A pickup doesn't distort.
Highly...highly...highly disagree. And the fact that there are 1000s of aftermarket pickups with all kinds of specific tunings (created by wire thickness, choice of magnets, etc.) supports that.
Otherwise, we'd all be using the exact same sounding PAF, right? And a Super Distortion from Dimarzio wasn't JUST created for more output....
But for funsies, try a Dimarzio ToneZone or a Duncan Invader in a mahogany Les Paul and then try a Norton or a Duncan Distortion in the same guitar. If the low-end response sounded the same to you in terms of tightness with all 4 pickups...well... oof.


Besides, we're not talking about pickup distortion here. Just frequency and transient response. I'm well aware that a beefy sounding pickup may overload a pre-amp and create a sluggish amp's response, but the differences I'm talking about are even audible when playing them into Fendery clean amps.
EDIT: check Mike Stamper's YT channel; he's done tons of pickup A/B comparisons:
https://www.youtube.com/@mikestamper/videos
Here, from 3m06s onwards you can hear the low-end response difference between the Invader and SH-6 Distortion with the same amp settings. And around 3m54s, when he starts the faster galloping riffs, you can hear the Invader is becoming a tad woofy, while the Distortion retains its clarity better.
And here, from a different Mike, at 1m18s you can hear their different responses when played clean. The SH-6 ain't no slouch in the low-end, but the Invader sounds bigger, but 'slower' too.