Is it possible to update Kirby within the panel?
If not, would it be a good idea? Or would we end up with clients updating Kirby, not testing the website, and ending up with broken websites?
You are basically already giving the answer why we decided against it 
Most of the times, some sort of double-checking from the developer or additional plugin upgrades are needed. That’s why we never felt like it’s a great idea to add this to the panel.
Also a ‘missing’ Update Button makes Kirby inherently more secure (compared to WP).
I explain to clients, that Kirby does not have a Core function to pull Files from 3rd party sources and let them run in production.