The e-ink display is a meta-stable device, it will show what was last written to it even when removed from the device and put away.
The fact that it never changes may be significant.
The fact that the charge controller isn't working (always yellow) and/or the battery is never re-charged is significant.
The fact that the presses on the power button cause no changes is significant.
The fact that it is an older device, which has seen heavy use is significant.
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hmm....
As described, it sounds like a double failure (rare, but possible) -
A broken power switch and a worn out battery.
I would replace the battery first,
but then not be surprised if I found that the power switch was also broken.
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