Hey, I'm an old dude with two SX basses and only one Fender and it ain't "old" it's MIM.
And I did just listen to the clips. And I don't have "golden ears" and I doubt that I could have picked the SX out without any hint. But once I HAD the hint the validity of my comments on pickup swap become obvious...or did they?
What my ears heard was that both basses sounded VERY good. But what I would have been looking for had the pickups been factory would have been that slightly more "edgy" sound of SX factory pups. I love that sound so there's a chance I would have recognized it. Personally I have no idea what Lollar pickups sound like so how could I pick them out?
But now here's what I found on listening. The Fender sounded exactly like... well... like you expect a Fender to sound like. That standard "old school" sound that we've heard so many times. The SX in spite of the different pickups STILL had that slightly more "modern" tone. Not hideously modern mind you, just a tiny "edge" to it that I really like. And let me say these are really minor differences I'm talking about! But I find it interesting that the SX bass sort of kept it's tone even with the pickups swapped. Or maybe it's just that lollar pups sound kind of like SX pups?
So my bottom line is (no surprise) I like the "modern" edge of the SX has best. It sort of speaks to the listener that this is a recent recording and not one from the vault. On the other hand if you want to exactly duplicate all that old stuff out there, then spend the cash. Clearly an old Fender still sounds exactly as it did when the old recordings were made!
As for how a bass feels when you play it, that is a personal issue. I hated the feel of "old" Fender basses when I played them back when they were "new" Fender basses. But that is a different issue from tone.
A good portion of what you are describing about the SX , I would attribute to fresh out of the pack strings that didn't get "played in" a little to take the edge off for that particular session.
The more "modern" sounding bass in my fleet, is the 2008 American Standard actually.
) SX as compared to a '64 P.