Leave - He would be a bigger star in other promotions

When guys like Jack Swagger, Damien Sandow and Drew McIntyre all left WWE at various points, they were all treated like bigger stars in their new promotions.
Sandow was pushed to a secondary title in Impact before his character became some weird Liberace knock-off.
R-Truth wanted to 'kill' former WWE writer. More details HERE
Swagger is the current Lucha Underground Champion after season four and McIntyre was treated like a big deal during his run in Impact Wrestling. His (McIntyre's) run in Impact directly led to him being resigned by WWE.
Regardless of where he might land (MLW, ROH, Impact, AEW), Ambrose would be one of the top stars in that company. AEW will need superstars who are credible as main-event superstars, and unfortunately for Cody Rhodes, he doesn't check that box.
He might have won the ROH Championship, but his booking in New Japan was that of a secondary champion.
Should both Kenny Omega and Ambrose sign with AEW, they'd already have two main-event stars that could help propel the promotion forward. It would also allow them to build up superstars who might not be 100% ready for the main event.
While Hangman Page might be getting there, he's still scratching the surface of being a main-event star and it would scream of nepotism in a way if he's booked like the top dog.
If you want fans to believe in the credibility of your new promotion, then an outside star like Ambrose would give you that if made the champion.