I would argue that async/await is far more important for native applications. I've seen plenty of apps doing significant work on the UI thread and hanging. Making it easy for developers to run things in the background is great for performance. Web browsers automatically add a level of asynchronous interaction.

It's also important on a web server but it won't improve performance for a single user in isolation. As you say, it allows more requests to be serviced concurrently. This won't speed up an individual request unless the server is already heavily loaded.

Edit: You don't need to build two library APIs. You can just build an Async method as the default implementation then have a blocking call to that as the synchronous method. I believe that all new MS APIs are now Async by default.