Theme settings only support general (global) styles that change in all affected elements.
I also support this requirement - to be able to set the button color separately from the general style in the theme template settings. This is because it is often necessary to visually highlight a button in each specific case (on a record page, in a pop-up window, etc.). Highlighting a button with color helps to visually identify a specific situation with a workflow or action - it tells the user that actions can be different (for example, if there are several buttons and they are responsible for separate processes).
For now, just make a static (constant) select of button color. Ideally (in the future), make a dynamic color select (using conditional rules based on variables and values from related collections).