You are correct. Slow blocking codes can cause poor performances both on the client-side as well as server-side. You can run slow work on a separate thread and that might work well enough on the client-side but may not help on the server-side. Having blocking methods in the server can diminish the overall performance of the server because it can lead to a situation where your server has a large no of threads running and all blocked. So, even simple request might end up taking a long time. It is better to use asynchronous APIs if they are available for slow running tasks just like the situation you are in. (Note: even if the asynchronous operations are not available, you can implement one by implementing a custom awaiter class) This is better for the clients as well as servers. The main point of asynchronous code is to reduce the no of threads. Because servers can have larger no of requests in progress simultaneously because reducing no of threads to handle a particular no of clients can improve scalability.
If you dont need to have more control over the threads or the thread-pool you can go with asynchronous approach.
Also, each thread takes 1 MB space on the heap. So, asynchronous methods will definitely help reduce memory usage. However, I think the nature of the work you have described here is going to take pretty much the same amount of time in multi-threaded as well as asynchronous approach.