what is the process called where a company seeks to match the benefits of a successful position while maintaining its existing position in offering customers a variety of differing services?

The process you’re asking about is called straddling.

In operations and strategy terms, a firm is said to be straddling when it tries to match the benefits of a successful competitive position while still keeping its own existing position and range of services. This usually means the company is attempting to enjoy advantages from two different strategic positions at once (for example, low cost and high variety), rather than fully committing to one clear, focused position.