I always wonder how much of the "vintage" sound has to do with the recording equipment and technique of that era, any idea somebody?
The signal chain remains somewhat simple. A bit of EQ, maybe a little scoop in the tone department. A lot of that vintage tone comes from a DI into the board, with maybe a bass cabinet mic'ed and mixed in for a bit of presence. Compression doesn't hurt either. I find it's easier to leave the tone up on the bass and adjust at the amp, unless you're planning on an overdriven sound, which then you should fiddle with your tone accordingly to the sound you're looking for.
Also, by the way, you can get that vintage tone with a Jazz bass, Hofner, Ric... it's all in the strings and the way you play 'em. Labellas, TIs, well broken in Chromes and Rotosounds, but even Fenders and especially GHS strings. Also, if you want to pay for extra coin, I also recommend the Optima flats. Great sounding set with a nice clickiness at the top end, and they're very responsive to picking too.
If the guys in the videos would play closer to the fretboard, they'd get an even fatter tone when picking. There's also the economy of movement to consider. Pick too hard with too much of the pick, and you'll lose some bottom end as well.
Finally, foam, foam, foam at the bridge. Not very much, and not too dense. Scoops and filters out overtones that muddy up the sound a little.