Easy. You can get '60s tone dead on with a good Road Worn Jazz through a DI strung with Fender or Labella flats. Skip the mutes. Even broken-in rounds do it. A used Road Worn Jazz will cost around $700 or less.
As far as "player as a tone factor" is concerned, this may be a bit over-estimated, IMHO, in comparison to the effect of the '60s studio technology and the studio/label system itself. Most of the hits in the '63-73 "'60s decade" were played by a handful of studio players, not all of whom used a Jazz.
In the US with a Fender Jazz Bass, I think Harvey "Brooks" Goldstein, Jim Fielder, Charlie McCoy, Chris Hillman, Joe Osborn, Norbert Putnam, and Carol Kaye literally set the tone. One wonders how much the major LA, New York, and Nashville cats studio tone became the tone in our heads for my generation.