So to actually answer your question, I'd say usually a streamer from an online service should not sound better (usually). Assuming your external hard drive is good quality and you have no equipment issues, and the files on your HDD are full resolution, then your DAC will deliver CD quality music. Frankly, most music servers are the same setup except the hard drive is inside.
Streamers are so expensive and work hard to not drop any bits to ensure they provide "CD quality".
When I first started streaming, I compared it to a CD and found that the streaming service sounded better....then I realized I was using a CD player instead of a CD transport feeding my DAC. So what I was really comparing was the cheap DAC in the CD player to my high end DAC seperate. Apples and oranges.
The exception I mentioned is music that is now available on streaming services that is higher resolution than CD quality. It takes a good ear and a good system to tell the difference, but it is there.
Like previous responses, I prefer the endless library and convenience of a streaming service.
Jerry