I don't get the breaking and re-establishing IC thing, this has nothing to do with power issues.
Sns, there are two possible relations that I can envision, albeit somewhat vaguely:
-- If a ground is missing somewhere, a component by itself (not connected to anything else except power) would tend to have its chassis and signal ground "float" to a voltage level that is determined by small leakage currents in its power transformer, or elsewhere in its internal circuitry. Connection to other components via interconnects would change that, because the interconnect shields tie together the chassis (and consequently the signal grounds) of the connected components. If multiple components are not properly grounded, they will "want" to float to different levels, but the interconnects between them will prevent that. Temporarily breaking the interconnections will allow the chassis of each component to return to the level that is determined its own internal leakage paths.
-- If the proper ground paths and ac return paths are not present (especially if ac neutral is open somewhere), return currents could be flowing through roundabout paths that may include or be affected by the presence of the interconnects. That conceivably could be causing something, such as a power transformer, to overheat and cause the audible symptoms after a warmup period. For instance, if ac neutral is open, the leakage path through a power transformer to ac safety ground could be breaking down and serving as part of the return path (for a component with a 3-prong plug), which would definitely overheat the transformer. For a component with a 2-prong plug, the return path in the absence of ac neutral could be via interconnect shields to other components and through them to safety ground.
Or something like that; I obviously can't formulate all of this precisely, but my point is that a relation between disconnecting the interconnects and a power problem is very conceivable.
Good point about the overvoltage possibility. Re your suggestions about multiple dedicated lines, etc., those are all excellent suggestions for optimizing sound quality, but I think that what we are dealing with here is a basic functionality and reliability problem, which needs to be resolved first.
Regards,
-- Al