[FEATURE] Log telemetry events and configuration sources

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

I often need to setup telemetry via the managed-settings.json file and in some occasions, there are setups that don't seem to send any metrics. My backend does not seem to be receiving any metrics from those users either, so it seems like it's either a Claude Code setup problem or maybe a network problem (maybe a proxy?)

I currently feel in the dark when debugging this issues, because most of the times this works fine but there's some individual cases that are hard to debug.

Proposed Solution

I'd like have a log entry showing if the settings were properly picked up from the managed settings file (or any other source) and show events of when metrics are being sent to the designated backend. If there's an error while sending those metrics, I'd also like to see them so I can correctly diagnose the problem.

There could also be an initial log showing the config so that I can validate Claude Code has correctly picked it up.

Alternative Solutions

I haven't been able to fix these issues because I don't have a good way to debug what is happening.

Priority

High - Significant impact on productivity

Feature Category

Configuration and settings

Use Case Example

I'd use this feature anytime a client reports that their metrics are not being tracked. After an initial check to validate that the managed settings file exists (and is correct) I'd go to the logs to check if there's a problem on the Claude Code side.

Additional Context

_No response_