How are small triacs able to “switch” or control high power AC loads
As @TonyEE pointed out, the hFE is on the order of 1600, which is really the product of the hFE values for the two transistors that make up the TRIAC. This gives an average value of just 40 for each, which is not unusual for medium-gain power transistors in saturation. 

If you read the full spec sheet you'll see that certain conditions apply. For example, you need to use some method of heatsinking to ensure that the case temperature doesn't exceed 86C if you want the part to be able to conduct 16A continuously. And if you're looking to switch a 35A load, you might want to look into the BA40/BA41 devices.

But you don't need to supply 4A to the gate the 4A value is the maximum the gate can handle without being destroyed (and even then it's only for a 20us pulse). Chances are pretty good that you won't be providing this amount of drive current with standard logic-level devices.