Admittedly, such a button would be nice. But it would also imply bestowing the software with the capability of changing its own code and/or files. That would open an attack-surface. ven with proper coding, it would be a constant weak spot.

Instead, a console based PHP file returning 0 or 1 if an update of your passwork instance is available would be more helpful. Then the admin can code a cronjob to perform the git pull, and possible backup jobs preceding that. Bash is pretty flexible.