I think that Overlapped I/O was invented to be used on the server side, mainly to be able to handle thousands of clients. So I do not think that there is any reason to use it on the client side. is my assumption correct?
Are you assuming that a client only needs 1 connection at a time? A "simple" browser could have 10 pages open, downloading 10 files, playing 10 videos, etc. Overlapped I/O would be a great way for a client to remain "responsive".