Yeah, trying to change it manually in the stock web browser is a real good way to break that browser.

The current official version of Opera does not have the browser agent setting that was present in previous beta builds.

The current version of Firefox for the N9 in the Nokia store does let you set your browser type and it works well if you install the Phoney add on.

Setting Firefox to "Firefox Desktop" mode results in the full desktop versions of both youtube and facebook perfectly.