There's no direct API or built-in export functionality for the content within the pages themselves, especially if it's rendered within HTML embed gadgets.

Instead of embedding HTML tables directly, store your data in Google Sheets. Use Apps Script to dynamically pull this data into your Google Sites pages. Embed your data as JSON within tags on your pages. You can then use an Apps Script web app to crawl your Sites pages, parse the JSON, and send it to BigQuery.