Question

what idea does the shifting of spectral emissions of distant galaxies toward the red end of the spectrum support

Answer 1

Observation is never used to 'support' a hypothesis or theory.
The hypothesis or theory is a suggested way to explain what
has been observed.

The expanding universe idea has been offered as an explanation
for the observed red-shift of the radiation from distant galaxies.
To this day, it's still the only way we have of explaining what we see.