Yes indeed, Fender's MIA P-bass pickups are tame, with a soft attack, but they sound amazing with Chromes. I had exactly the same experience when I used Chromes on a project P-bass with the pickup from my 2004 American Series P. (That bass now has a Lindy Fralin pickup, which is fantastic.)
However, I think the issue is partly the modern-voiced cabs with tens, though which I think a Jazz would likely sound better. I used to have a Goliath III, and my Jazz simply crushed my Precision through that cab, because the voicing allowed the J-bass growl to just rip through the mix. And it wasn't exactly a thump-monster rig.
Through more old-school rigs, especially with fifteens, very different story. The more old-school, the better the P. In fact, through one such rig at a rehearsal studio, my J sounded anemic in comparison to my P.
I now use G-K Neo112 cabs, which have enough booty for a P-bass to thump nicely, much better than through the Goliath. But even so, I think the modern, full-range voicing is slightly better suited for my J-basses.
Of course, I still love a P-bass through modern rigs. But, based on my experience, I could understand why someone using those rigs might feel a letdown going from a J-bass to a P. Maybe try dialing out some high mids and treble, and boost the low mids a little.