I agree with your statement. This is how I would assess probabilities as well. I am trying to keep an open mind, and look at the evidence. And so far I don't see how this can be explained by a failed component.
Yes, the schematic for E3646A seems to match E3640A (if you ignore the second channel and mismatch of the numbering of designators).
There is a DAC and 4-way multiplexer with precision caps for sample-and-hold and TL074 buffers: voltage ref, current limit ref, overvoltage protection ref and one more ref, which is used as some sort of an adjustable base line for the other three. Everything there seems to work as expected in general, but the voltage ref is sitting a bit higher at power up, but drops as soon as the output is enabled. And it stays there when the output is disabled. So the DAC must be fine since it is shared for all refs, the multiplexer seems to be fine too. The only explanation I have at the moment is that a wrong value is sent to the DAC after power up, and it gets overridden by the right value once the output is turned on. I wish I had some sort of a protocol analyzer to capture the data going to the DAC. Perhaps it is time to get myself one of those inexpensive USB logic analyzers. But suppose I confirm this hypothesis. What is to be done? Can it be a corrupt parameter in the EEPROM (93C66)? Is there a way to fix that at all (copy from a working unit and recalibrate)? Maybe it is too early to worry about that. I need to check that somehow or come up with some other explanation.