The "API" you are referring to is known as SQL, transmitted over the wire using the pgsql protocol. There's nothing to copy. The magic of Cockroach is in the implementation.
Yes, it is indeed a little puzzling. CrB's biggest benefit is to people not interested in engineering their own scaling/sharing solutions for existing RDBMS options or if they care about optimal resource utilization (under a certain subset of conditions), neither of which are particularly draws for the AWS team. Perhaps they are more worried about other vendors.
How much a SaaS has to do to not be consider as selling CockroachDB as is?
These questions apart is worth noting that AWS can still copy CockroachDB APIs, rename it AWS Ants, and called it a day.