They're getting the info directly from Google et al., but they don't have root on Google's servers. Google is required by law (CALEA, the Communications Assistance for Law Enforcement Act) to provide the ability for law enforcement to get information from them. This includes - required by law - the ability both to get stored data and to make real-time intercepts of new communications. Google is paid a fee to provide these services as well.
Google et al. have fully complied with this law. The FBI manages the government-end of the CALEA tapping capabilities. The NSA makes requests to the FBI, which passes them on to Google, which flips a switch and enables the tapping of user "xyzzy123". From then on, xyzzy123's stored data and new communications get sent to the FBI through the CALEA connection, which forwards them to the NSA. CALEA also requires the service provider to provide all sorts of metadata about the user.
This IS "direct access" to Google's servers. The denials about this have been carefully worded things that all access is supported by some sort of legal process, etc. The denials are non-denial denials. Yes, GOOGLE (et al.), not the NSA, flips the final switch which sends the data. But Google is required by law to do so, so....... And once the switch is flipped, all of the data is flowing automatically to the NSA.
I hope this is clear.