Singer HE-2200 Kids Record Player

You said the magic word - Germanium!  Odds are these caps have long since died.  The transistors back in the 60's were not cheap, and cheap ones were really, really cheap garbage compared to today inexpensive transistors.   

Assuming you have a PS that works in the thing, then take a look for burned out parts.  I'm guessing there are none, given you have Germanium tranny's in there.

I suggest you replace the transistors with a basic NPN transistor, assuming the germaniums are NPN style.  Try a MPSA42, which is a good audio transistor and rated for 500mA and 300Vdc.  The bias should be fairly close but may not be close enough.  If not, you can play with the bias resistors a bit, the A42's will take less bias. 

If you are really ambitious you can just build an OpAmp circuit and stick that in it.  A LF356 is a good choice and is fairly benign to its surroundings.  You might need a current buffer to get the power you need.  

A better option is sticking in a power op amp, such as a UA7307TC.  This part is just about made for this application.  You can get all these parts at DigiKey.com along with datasheets and app notes.

But barring that bit of good luck I’m hoping someone can suggest a very simple (and cheap!) amp module that I can put in. It would need to work with a ceramic cartridge.

You can find a LM386 amplifier module at DigiKey.com, however, you still need to add an emitter-follower in front for the ceramic pickup.

I'm going to look in to what can be done with the amp.  I misspoke above, it is actually 4 transistors:  2 2SB54 one after the other on the input end and 2 2SB56X2 paralleled driving the output transformer, one driving each end of the center-tapped transformer input winding.  (I think all are Toshiba.  Maybe they made the entire little amp.)  I cannot even find specs on the output transistors but given the static output I'm guessing they might be okay.  The preamp stage transistors are available.

I may go ahead and unsolder those two and see if I can make a rudimentary determination--perhaps look for the diode effect.  I will probe around to see what I can find on the resistors and unsolder for a proper test if anything is suspicious.  60 year old electrolytics should probably be replaced but we'll see.

The volume control pot is actually on the input end.  Pot is supposed to be 1M and measures nearly double that end to end but the wiper seems to be okay and ramps up 0 to 1.9.  I don't know if the end to end resistance is an issue as one end is just to common.

The phonograph runs on 6V provided by 4 D cells.  One side is directly common to one side of the speaker!