If you would like to center all tabular environments, I would recommend to create your own environment with a different name, which calls tabular. You could use that instead of redefining tabular itself.
But no problem, this can be done. Just use
\renewenvironment, for example:
This as addition to your first try redefining tabular.
You could redefine the floating table environment very similar:
You could also use etoolbox to patch the table environment or append
\centering to the definition this way: