According to MDN, Chrome ignores run_at for CSS: "In Chrome, runAt does not affect the CSS insertion point."

See mdn/content#14725 (comment)

Chrome docs have conflicting info:

• extensionTypes/RunAt: "The soonest that the JavaScript or CSS will be injected into the tab."
• content-scripts/run_time: "The run_at field controls when JavaScript files are injected into the web page."
• content-scripts/css: "These are injected in the order they appear in this array, before any DOM is constructed or displayed for the page"

According to MDN, Chrome ignores run_at for CSS

I don’t think it's quite accurate: CSS must always precede the JS and it should precede the first paint, as implied by the table:

document_start string Scripts are injected after any files from css, but before any other DOM is constructed or any other script is run.

My expectations are:

• The CSS is injected at the end of head, unless run_at === document_start, in which case it's injected at the beginning of head

This would explain what you said before:

CSS specified in the manifest overrides page inline styles in Firefox but not in Chrome

Anyway, easier to test than investigate