This is from the log file located at /Users/-USERNAME-/Library/Application Support/Battle.net/Logs:
D 2024-08-17 08:52:16.880724 [InstallManager] {Main} Game is running: wow
D 2024-08-17 08:52:16.888699 [ProductState] {Main} InstallState (wow): gameRunning=1 isGameProcessRunning=1 GameLaunching=0
I 2024-08-17 08:52:19.893509 [GameSessionList] {Agent} Reporting process: TrackingInfo(uid=wow, type=game, opStatus=Off, reqId=0, running=0, oldRunning=1)
I 2024-08-17 08:52:19.899839 [InstallManager] {Main} Setting Process Running: false uid=wow binaryType=game any=false
D 2024-08-17 08:52:19.899901 [InstallInfo] {Main} Operation status changed: opType=Execute oldStatus=On newStatus=Off agentUid=wow
D 2024-08-17 08:52:19.899919 [InstallManager] {Main} Game is no longer running: wow
Seems like the executable crashes and removes itself from the running processes in 3 seconds 
From a fresh test:
I 2024-08-17 09:01:08.469516 [InstallManager] {Main} Launched /Applications/World of Warcraft/retail/World of Warcraft.app with args: -launcherlogin -uid wow, pid: -1
I 2024-08-17 09:01:08.469558 [InstallManager] {Main} Failed to launch executable, syncing game data: agentUid=wow
D 2024-08-17 09:01:13.363641 [InstallManager] {Main} Game finished versioning wow
E 2024-08-17 09:01:23.253410 [InstallManager] {Main} Operation ‘Execute’ was pending and expired. uid=wow