fix: make updater work with private repos

[sitek94]

Hey, first of all: great repos - both ChatGPT and this one!

I noticed small issue with updater script when using it with private repos. Namely, it produces "Not Found" signatures, e.g.

The issue turned out to be using browser_download_url to fetch signature. Even when I passed token in headers, it still wouldn't find the file.

It can be solved by using GitHub API url: https://api.github.com/repos/${owner}/${repo}/releases/assets/${assetId}

Also, I got some TypeScript issues when cloned the repo, so I fixed those as well.

Cheers