Messages about an operation's success, failure, or progress are known as "system status indicators". Jacob Nielsen included them in his 10 usability heuristics as far back as 1995:
Visibility of system status
The system should always keep users informed about what is going on, through appropriate feedback within reasonable time.
Thus, discarding the progress does indeed hurt the usability of the site.
BEWARE: the Nielsen heuristic should not be taken literally with a toast/flash message for every time an operation succeeds. Changing the state of the control to what a user expects is also considered to be a system status indicator (e.g., flipping a light switch turns the lights on/off or an interface's color change is applied instantly). Overabundance of notifications can be distracting, so please be careful and test with real users to achieve a balance between too much and too little information.
