It actually does everything you propose.
For render props, the BaseWeb API is not <Select renderDropdown={...} renderArrow={...} renderLabel={...} /> but it's collapsed under a single prop overrides={{ Dropdown: { component: {...} } ... }} naming convention. This "map" includes (and gives you access) to every single subcomponent.
The same goes for "adapting styles based on props".
BaseWeb uses exactly same patterns and techniques that you mention. The difference is the naming consistency and uniformity across all components. Instead of "littering" the top level API, we have a single prop "overrides" and instead of deciding what should and shouldn't be customizable, everything is customizable. Again, without a need to keep adding new top level props.