Revamp logging to be more useful and easily filterable

### Is your feature request related to a problem?

The current logging in private-mirrors is inconsistently implemented and hard to filter to useful traffic for monitoring a production deployment.

### Describe the solution you'd like

The logging should be updated to ensure that appropriate log levels are used for all scenarios, console.log is never used, and all potentially sensitive information is masked.

### Describe alternatives you've considered

It may be worth switching to a different logging library like Pino which is one of the fastest and most popular logging libraries for Node development.

### Additional context

_No response_