Yeah, the white is embedded in the sidewall on ONE side of a tire and exposed by grinding off the black cover rubber. But that only happens on tires that have white - WSW (White Sidewall), RWL (Raised White Letters), OWL (Outline White Letters), etc. BSW (Black Sidewall) tires do NOT have unexposed white layers (except by mistake!)
Yes, carbon black is a common reinforcement because it is chemically compatible with hydrocarbons, such as rubber and it is abrasion resistant, making the tire wear longer.
Titanium dioxide is used to make the rubber white and it can be used as a reinforcement material, but it is not as effective as carbon black.
And, yes, the natural color of rubber is a translucent amber, but changes color with even a small amount of other materials.