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.
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.