Display API rate limiting details in health check

### Feature summary

Display GitHub API rate limiting (gh api rate_limit) details in the health check

### What problem are you trying to solve?

I hit GitHub API rate limits today which completely stopped me from working completely. The GitHub status page showed everything operational, but all my sessions were blocked due to "GitHub authorization needed". The built-in health check in the app reported that authorization was good for my account, which is weird. Trying to re-authorize my account always resulted in "failed to fetch user info after auth". I rebooted, same problem. I looked at the logs, and then saw that the call to fetch user info had failed due to API rate limiting.

If API rate limiting information had been displayed in the health check, I wouldn't have wasted time chasing the wrong problem. The rate limit info also includes the time at which the rate limit resets, so it's very useful to know if I just need to take a coffee break and for how long. It would also be useful to let me see if I'm using way too much API calls when I'm not supposed to.

### Proposed solution

Simply display the output from the gh api rate_limit command in the health check, that would certainly do the trick.

### Workflow impact

Critical - it stopped all my sessions and I couldn't get anything done.

### Installation context

_No response_

### Additional context

Start at the app's built-in health check and trace how GitHub authorization status is collected; use the gh api rate_limit output described in the issue as the reference. Done means the health check displays rate-limit usage and reset-time details alongside authorization status, making limits visible when they explain failed GitHub requests.

索引モデルが issue の本文から書いたものです。