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    Building, troubleshooting and testing of these amplifiers should only be
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12AX7 for a cathode follower?

more appropriate
For what? Driving 500k pots? Driving 833 grids? Driving modern studio inputs?

Why ask? Wire it up. There's at least a dozen tube-types which will plug in and work "the same", see what suits you.

FWIW: in the 'audio' range, the AX7 may be the most used cathode follower in serious gear, going back to the K2-W opamp.
 
It's at the output of a freestanding phono stage and will drive the usual RCA inputs of the next box.

Happy to change passives around the socket to optimize, but I can't change to a different socket.

It's more or less based on Marantz Model 7 phono stage, which uses a 12AX7 for this. Thank you.
 
Hmmmm... I am no expert but my thinking is that an 12AX7 would be fine to drive a feedback network (say a 12AX7 gain stage direct coupled to a 12AX7 CF fed back to 1st stage grid) but if driving an external connection why not switch to a 12AT7 (with appropriate cathode resistor change) to give better drive in case you decide to use it to drive a solid state line amp?