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I agree that's one way to look at it. I prefer the framing that the signal just isn't that interesting to listen to.

Compare to the cosmic microwave background - we have no problem converting microwaves to visible light (implicitly doing a frequency shift), but the picture you see doesn't vary despite us having had enough time to collect many cycles of data.

The gravitational background should be similar - even if we collected data for thousands of years we're not going to hear a cool 'bloop' like for black hole mergers because interesting transients just aren't in the signal.