There is no “Arrived Button” in a tower.
What is happening here is a function of the ATC radar systems ( ARTS). When an aircraft gets close to an airport at a fairly low altitude the data block will disappear. ( It will appear on the controllers “Coast/Suspend” list). The controllers can restart the track in the event of a go-around, or the computer will recognize the transponder and restart the track. It is the ARTS computer that marks the aircraft “arrived” ( there are lots of different parameters for this).
I think the correct question here is: How does the data feed used by Flightaware pick up tracks that may have been in the Coast/Suspend list, especially in the case of a Go-Around?
The Sky West RJ is a perfect example. The aircraft descended and went around. The track might have went to the Coast/Suspend list. After the Go-Around, the track was re-started, and the aircraft headed for SBA. The departure controllers worked their magic and entered a route to SBA… LAX255015…SBA.
So, the controller never lost a data block. I’m guessing the FAA feed that Flightaware gets doesn’t have anything until the new flight plan data is entered, and then the tracking continues for Flightaware users.
On a lighter note: If controllers had to use an “Arrived Button” the controller handbook would have an entire chapter devoted to when or when not to press the button, who to call when the button was or was not pushed, appropriate strip marking for the button status, and which position in the tower should be responsible for the button. And an entire new ATC glossary entry.