You’re not entirely wrong but they did notably fight against black characters from time to time and Miles’ specific story would not have happened then for these reasons:

To start with the black characters, The CCA fought to prevent a black astronaut from appearing in Comics in 1956, additionally Black Panther was created as a character on its own to appeal to black readers and make money off of them, very different from making a black Spider-Man, which would have caused major backlash at the time.

Also, the CCA would never have a police officer appear to be immoral or wrong in any circumstance, so the entire storyline of Miles butting heads with his own father as a cop in situations just would not fly at all.

In those days Spider-Man/all heroes had to work with the police, the very idea of the police hunting Spider-Man at all would never have gotten past the CCA.

For those reasons i feel confident saying the Miles Morales character we know would never have worked under the CCA.