The outbox pattern only works inside an NServiceBus Handler. However, for NoSQL DBs like Cosmos (and perhaps RavenDB), it would be great to get it working with ASP.NET Core Web API.
Propose scenario:
We save the data synchronously in Cosmos DB when calling a Web API along with events that need to be published to an outbox document in a single transaction batch.
A separate process/ worker reads the data from the outbox document and publishes the event.
Advantages of this approach
Saving the data in a separate process through a handler has its own complexity as we need to use additional services such as SignalR to notify the client when the data is saved. Hence, it would simplify most of the use cases since the save to DB is synchronous.