The Wikipedia article on the subject seems to lean heavily on red shift as the sole observable basis for the claimed expansion:
https://en.wikipedia.org/wiki/Expansion_of_the_universe
The other “proofs” appear to be mathematical and not empirical.
In addition the claim that gravity has local effects but that does not stop expansion at large scales appears to be in conflict with Einstein conceptions of gravity (since of course curved space time is everywhere under the theory).
Imho modern physics is busy pounding square pegs into round holes. Claims of poorly defined (and not empirically verified) “dark matter” and “dark energy” look ripe for being overturned by future observations and theory.
(That said I do not claim to have a solution to these issues.)
One of the additional evidences for expansion is the delayed light curve of distant type 1a supernovae. Type 1a are standard candles—that is they are all about the same brightness and their brightness fades at about the same rate after the initial explosion. But the farther away we see them (their distances also indicated by Cepheid variables in the same galaxy), the slower their brightness fades. This is because Special Relativistic effects are occurring due to their receding away at speeds also indicated by their red shift.
Another one is the higher CMBR temperatures observed in distant intergalactic gas. As the universe expands, the CMBR cools, as predicted.
There are a few others, including observed evolution of galaxies with distance and the increasing abundance of quasars. All these point to expansion. If the red-shifting of distant galaxies are due to something else, it is not known what. The idea of “tired light” has no supporting evidence at all.