After already parting ways with veteran forwards Jesse Hogan and Jake Stringer, the Giants are about to lose the greatest player in the club's 15-year history.
A member of the inaugural GWS team back in 2012, Greene is moving back to his home state Victoria for more family support for his young clan.
The Giants' list overhaul might not end there, with the club to consider offers for experienced contracted pair Lachie Whitfield and Harry Himmelberg.
Matthews, who will leave his role in the next few weeks, made the extraordinary step of announcing Greene's decision to leave on radio on Thursday.
He was also behind GWS famously revealing superstar Lance Franklin had chosen to sign with Sydney instead of the Giants back in 2013.
"It's not unusual for us to have some changes on an annual basis to the list," Matthews told SEN.
"Clearly, this is the most high-profile and significant because he (Greene) has been captain of the club.
"There's a bit of a reset, but it's not like it's a rebuild or there's any sort of major issue from my point of view.
"The club's in a more stable position than what sometimes gets presented."
Only attracting an average home crowd of 11,531 this season, the Giants can ill-afford to go through a lengthy rebuild.
Capturing an audience and building support in western Sydney is a long-term problem for the AFL.
Greene's looming trade leaves Stephen Coniglio as the player left from the Giants' debut season.
Coniglio is not expected to join the exodus, instead staying on for a 16th season with the Giants.
"This decision hasn't come out of the blue for us, we were pretty much expecting this might be the case," Matthews said of Greene.
"We'll celebrate him accordingly.
"But the club is bigger than any individual, and the club will move on.
"The club, members and supporters have got every reason to have confidence in what's ahead."
In a tongue-in-cheek social media post, the Giants referenced classic US sitcom The Office with the song, "Goodbye, Toby''.
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