I'm not sure if blocking third party cookies is the cause, but I have had difficulty downloading files under Firefox. Disabling third party cookies is one of the first settings I enable after installing Firefox.
I would not be surprised if Apple makes an exception for Google.
No point disabling them now due to Total Cookie Protection by the way. All 3rd-party cookies are isolated to the 1st-party domain they're obtained from.
I would not be surprised if Apple makes an exception for Google.