If you are not familiar with the site, vintagehofner.co.uk has info on the older PUs with some good gut shots in the sub links:
Hofner Guitar Pickups
@markjazzbassist
Are you comparing the Höfner staples to Fender Jazz PUs?
HBs will cause some cancellations which SCs won't, but in general both are very capable of producing the same lower frequencies and not sound thin [depends on the wire gauge/winding count/magnet strength used in either] (think of a Höfner 500/1-2-5 with the simple Type 1 or blade single coil PUs, they can also do lower frequencies in spades).
The PU positions will be the main cause of the differences (aside from the different cancellations caused in dual PU mode with both). The Höfner 500s have the front PU mounted (like modern Rics or the old EB0s) just south of the 24th fret position, the Jazz front PU is much closer to the bridge, about 1.5" to the north of the 36th fret position (the 36th is where Ric and MM SRs have the back PU centered). If you shifted a Jazz front PU up to the 24th fret position, soloed you might find it closer to a Staple neck PU (or type 1/2/3/Staple or Bar) then you'd think.
The Gibson Sidewinder "mudbuckers" had two huge heavily wound coils and two 1/8" magnets which causes their unique "wooly" tone in that position. FilterTrons [Thunder Trons/Blades] use a 1/4" magnet, smaller coils &
much lower winding count, which allow them to retain more clarity top end and still have a solid bottom not usually associated with e.g. PAF HBs types (higher winding & with a single 1/8" magnet).
The different scale length & hollow body vs. solid & bridge types make a difference especially with the sustain, but the PU positions tend to cause the biggest differences (assuming the PUs are not drastically voiced like a "mudbucker").
This article is interesting, the poster is aiming for making a JB sound like a Ric - unfortunately w/out sound samples, but the same principle applies. There are a couple of threads here where similar projects were done, but this picture is good:
X-Bass project: 4 pickup Jazz bass chasing RIC tone. - MOD WIGGLER
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I have always had a hankering of picking up a used CN Hofner CT Club and doing something similar to this, but a slightly less invasive rout which the CN 500/2 top mount PUs would allow: