Built a feature
Wrote a Chrome Extension
Today I figured out which key APIs are needed in order to turn a Chrome extension into an automated scraper/text-inputter.

browser.tabs.onUpdated

That's what you're going to want when you load up a new page. It's an event listener in the background page that fires off when a page is going through its loading lifecycle. The loading status is accessible through 

changeInfo.status
changeInfo is available in the onUpdated callback function, and when that becomes "complete", you can send a message to your content script to fire off any scraping/scripting.

So a basic example would look like

//content-script
sendMessage('start-scrape', {link: link, nextMessage: 'find-elements'}))
onMessage('find-elements', () => {
    // this function will get triggered by the background page,
    // after the page is fully loaded.
    // so the scraping logic would go here
})


// background-script
onMessage('start-scrape', ({data} => {
    // this updates the URL of the currently active tab 
    browser.tabs.update({ url: data.link });
    // we name the function openPage so we can remove this event listener
    browser.tabs.onUpdated.addListener(async function openPage(
      tabId,
      changeInfo
    ) {
      // this event gets fired multiple times on a page load
      // so we only want to start scraping when loading is complete
      if (changeInfo.status == "complete") {
        browser.tabs.onUpdated.removeListener(openPage);
        sendMessage(
          data?.nextMessage,
          {},
          { context: "content-script", tabId: tabId }
        );
      }
    });
})
It was tempting to assume that I could just use some familiar APIs to check if the document was loaded, but the browser API's message-passing system and onUpdated / onCreated APIs are actually much more reliable when you need to know the page content is fully loaded before beginning more complex automations.

I look forward to being able to share some more progress as this prototype becomes a product.