Somehow I've got myself a "WTF" dilema with my newly setup Turntable. HELP!

Although some of the reported observations seem inconsistent with the possibility of a polarity reversal in one channel (which could conceivably result from a miswire in the new turntable, or even a miswire within the cartridge itself), to be certain that isn’t causing the problem I would suggest that you try reversing the + and - connections to one (and only one) speaker, at either the speaker end or the amp end of the cable (but not both).

As you may be aware, a polarity reversal on one channel can have strange effects on image position and focus when both channels are playing, but would not affect sonics significantly when only one channel is playing.

Regards,
-- Al

Sounds like the preamp/phono stage as others have stated. 
First level of contingency is backup so,  
I would sub in your backup preamp/phonostage to confirm.
Good luck

I propose you are confusing the setting for your cartridge.  You have a high output MC cartridge which will require the same gain setting as a MM cartridge.  So that means a low gain output setting.  

You refer to to a high output setting which is the exact opposite of what you want with a high output MC.  I looked up the spec for your phono stage and it provides a whopping 85 db on high and even a seriously high 60 db on the low setting.  Most phono stages provide 40 db low and 60 db high.

 I think you are overloading the input on the preamp.  Try the lowest gain setting.

Try the phone IC into Left of Aux and Right of Tuner. See if switching sources still plays audibly.
Trying to ascertain if the pre/power amp is loading the phono amp incorrectly - i.e. a phone amp issue. If it does play on one channel for both sources and not when both are in one source input, I'd suspect the phono amp.

Typically, the choice of gain setting is made at the input side of a phono stage.  Many/most that provide both MM and MC levels of gain provide corresponding "MM" and "MC" pairs of inputs.  This makes sense, because in nearly all cases, the major portion of the gain is created at the first gain stage after the input, so that the RIAA correction can be effected with a higher level signal than would otherwise be the case.  This is only to say that I would be surprised if the OP's phono stage has more than one pair of output jacks per channel, and if selection among those choices has an effect on phono gain.  But only the OP can say.

Among the hypotheses offered, I think the "channels out of phase with each other" explanation makes a lot of sense.  In that scenario, each channel would work fine if the other channel is not connected or not producing signal.  No doubt we will soon know whether that explanation holds water, because it's so easy to test.  Even more trivial explanations could work: (1) Internally broken IC that is creating an intermittent problem; (2) "Iffy" contact between the hot pin of one IC and either the output jack of the phono stage or the input jack of the integrated; etc.