Indeed a coupling capacitor isn't neccessarily acting as a capacitive load, although it may, in some cases. To know exactly, you have to calculate or measure the load impedance. A large capacitor can be treated as short AC-wise, so except for DC, the load behind it matters.
The sketched circuit surely can't work without a coupling capacitor, because the transformer is a DC short to the OP output. If the transformer is a suitable load for the OP - depends on. The winding capacitance may e.g. present a considerable capacitive load, or the inductance be too low to be driven by the OP. Also the circuit at the secondary must be taken into consideration. Adding a voltage follower would reduce the load to the OP output. If the follower can drive the load in the intended way is a different question.
I suggested LM6171 instead of LM7171 because it's unity gain stable and promises somewhat more circuit stability, if you don't need the LM7171's GBW. It has been mentioned also by other contributors, that these > 100 MHz bandwidth OPs are likely to design unwanted oscillators. Their requirements for low inductance ground and bypassing are considerably different from gentle general purpose devices. It's like driving a family or a racing car.