So then how do you manage the secret that authenticates an application's identity? And what good is the logging if after an application has the secret it can do whatever it wants with it?

if it is an instance on the cloud, GCP and AWS let you define ServiceAccounts that get populated on the Instance at boot time.

you should only let the instance access the secret it requires.

and how do you manage secrets that let you define that ServiceAccounts?

As OP wrote, you did not solve it, just moved it to a different level.