This is comparing apples and basketballs. The lurid arguments against the CRA were in fact from conservatives, but the parties weren't polarized the way they are now.

Before civil rights legislation, the Republicans were the party of capitalism, industry, protectionism, and to an extent urbanism; in the early 20th century they were defined mostly in terms of opposition to progressivism (antitrust, workers rights, regulation) and the New Deal. Democrats were the party of farmers, states rights, and (see previously) progressivism and the New Deal. You can look at this as a "conservative/liberal" split, but it's a different one than we have now.

It was only after the CRA that the process was set in motion to polarize the parties perfectly by conservative/liberal ideology (in the sense we use those terms today). And that process didn't even really finish until around the 1980 election; for example if you look at '70s Republican party platforms, for instance, they're still playing footsie with being pro-choice.

So it just doesn't mean much to rebut someone saying "conservatives opposed the CRA" with "no, the Democrats did". There were essentially 4 parties at the time: the liberal/conservative wings of both parties.

And let’s not forget these were very similar parties back then. Southern democrats were FDR democrats. Woodrow Wilson from 1916 is a model of a modern democrat that believes in experts and expert agencies running the country. George Wallace was a big labor guy.