Most crossovers in the 60's were called as a palming violation. It wasn't until the late 60's and early 70's that the palming rule started to give way as more playground influence was accepted into the "purists" (IE boring) Brown-Ball league (the NBA) because people started recognizing the entertainment value in the dazzling play being showcased in the ABA which encouraged more dunking/dribbling/individualistic play. Rules really got slack with ball handling coming into the AI era. The crossover move itself, and other moves we see today and think of as "modern" inventions had all probably been used on playgrounds for decades before they were used in the NBA. - and I have footage of a lot of moves like Eurosteps and crossovers being used in the 60's so that means these moves could have existed in the 40's on ww2 Aircraft carrier leagues and such for all I know. It likely just wasn't okay to use them in the NBA yet due to strict "purist" rules being pushed at the time.
For example, you had to dribble palms down. It's difficult to crossover when you can't put your hand on the side of the ball:
Why many 60s-70s NBA players don't crossover or display "handles" (stricter palming calls) - YouTube
Though it could be done, and was done, just less frequently:
Jerry West Crossovers - YouTube
Refs today let players put their hand literally underneath the ball as they dribble up the court. Ball handlers today touch the ball anywhere they want, they even are allowed a substantial amount of hesitation. Strict "Palms down" dribbling isn't in fashion anymore. The games rules have evolved for a more aesthetic and free flowing brand of basketball. But that doesn't mean fancy ball handlers today could go back in time and strut their stuff. They'd be committing all sorts of violations and turning the ball over. The crowd wouldn't go

rather, they'd go