Sorry, I think I meant transatlantic airport, or a "large" airport. Not sure, it used to lack something I heard but not sure exactly what and if it's still true. They're struggling with building a new one so something :)
I've just checked: there are plenty of direct flights each day from Berlin to New York for example. It is not the central hub like Frankfurt is, but I doubt you won't fly to Berlin because it does not have enough infrastructure...
The lack of direct flights is partly due to the short connection to Frankfurt from where you can connect to most destinations in the world. That's also why some other German cities (Hamburg, Cologne, Hannover) have few transcontinental flights, despite the demand. There has been an increase in direct flights from Berlin recently but towards the US, Heathrow and Frankfurt are on the way and BA/LH can price out competitors easily.
Berlin is certainly underserved given its size but that's just the Frankfurt effect (an airport with >60m passengers p.a. for a city <1m population).
There might be transatlantic flights from Tegel, I'm not sure, but they're rare. Moscow and the Canary Islands are more or less the limits. I usually find myself going through Frankfurt, Paris or Amsterdam for intercontinental travel.
I was in Tegel a while ago, flying home to Dublin, so was in a non-Schengen section (Ireland's not in Schengen). The flight at the next gate was going to Beijing. I was kind of surprised, because Tegel has a very "small regional airport" feel (partially due to the weird design), but apparently there are flights to the other side of the world from there.
Berlin has an international airport.