The APU is not on during flight, so not a safety thing. It's just for providing power (and bleed air) on the ground (if there's no other power source). E.g. can't start up at airports without external power with the APU INOP.
APUs have a common mode failure with main engines: lack of fuel. Are there designs out there which require emergency power after engine failure and don’t have a RAT?
There was a Mentour Pilot video on the Jeju Air crash that touched upon this. With a dual engine failure on a 737, you lose both generators and fall back to the emergency batteries (no RAT). The aircraft remains flyable, but it's extremely challenging - you need an operational APU to make things a lot more manageable: https://youtu.be/9GbmGUk8Y0M?t=2001
