What's the value in lots of capacitors inside a line stage preamp?

In a tube cap some of the most important caps are the coupling caps.  They block DC but allow the signal to go through.  Because all the signal goes through it their character can be the character of the preamp.

Second are power supply caps.  The higher the value, the lower ESR, and L the better they filter power supply noise.

CJ don’t use (smaller) electrolytic capacitors, hence all the film caps necessary for the required capacitance.

@noromance is also right.

I looked at a pic of the inside, and even going back to the PV10, CJ uses capacitors which are just enormous.  Film caps are larger than electrolytic but even among film capacitors there's large and then there's huge. 

Generally, a film cap's size has to do with the uF (microfarads) and the voltage rating.  It looks like CJ is using exotic film caps which for a given uF and voltage are unusually large.  Caps of Unusual Size.  COUS.  (Princess Bride Reference). 

It's also true, IMHO, that for tube preamps the capacitors are often THE sound of the preamp, so it's no wonder for me that CJ is going with exotics.

Not all capacitors store energy for transients. Those big ones in power supply - yes. They also act as part of filter reducing "ripples" in the DC which remain after AC/DC conversion.

Now, small caps in the amplifier are NOT for power storage. They serve many purposes. For example, isolating one cascade of amplification from another. Capacitor passes AC but not DC. Music is AC (not to be confused with AC mains). Another purpose is to improve stability at high frequencies. Some may not be even part of the amplification and rather operate buttons and such.

conrad-johnson (no caps on the names) have always used very simple circuits and very highly specified parts. Their reliance on film caps, including large value ones is contributes significantly to their product's cost.