Is there any way to bind a content script to Chrome's start page?
I tried setting matches to "*", but it doesn't even run. With "*://*/*" it does not bind.
No, you cannot*. Technically, the start page is chrome://newtab/, and Chrome Extensions cannot access chrome:// pages for security reasons, not even with the widest "<all_urls>" permission.
Your only hope is to make your own New Tab page, though it would be hard to replicate all of the default functionality (e.g. thumbnails of top sites).
* One can enable this with Chrome Flags: chrome://flags/#extensions-on-chrome-urls But this is only applicable if the extension is for personal use and is a potential security risk.