In original Planet of the Apes film, could it really be possible that the only remnant of the city would be the statue's head? Aren't there a lot of structures that would be more likely to survive all that time?
A big assumption is that the statue remained on her pedestal on the island, and even that the statue is intact. It's entirely possible that she was ripped from her moorings and deposited elsewhere nearby, either in whole, or in part.
But, let's work with that assumption for the moment. I think some confusion comes from the fact that many of the images we see of Lady Liberty show her with the famous NYC skyline directly behind her. However, that skyline is actually off to her left (observers right). Directly behind her is New Jersey!
Heston is facing in roughly the direction given by the red arrow below:
As you can see, not a lot to see there.
In the second movie of this series, it is shown that the mutants living in the old city have the mental capability to project illusions - recall, Taylor disappears by falling through what appears to be a solid stone wall. The towers of NYC are hidden behind these illusions - as is shown later, when the ape army gets past the illusions and sees the heavily eroded skyscrapers.
Beneath the Planet of the Apes - New York (in the year 3979):
Short answer: Yes, based on what is presented on-screen: It really is the only remnant. It is also the only remnant Taylor sees. I re-iterate: On-screen! We (the audience) have absolutely no way of knowing whether, in fact, there are also other remnants visible off-screen. For all we know, the Eiffel Tower might still be standing in Paris, and the Golden Gate Bridge might still be standing in San Francisco. That is the definitive answer about the appearance of remnants on-screen. If you have a question about what remnants might still be in existence elsewhere in the world off-screen - please submit a separate question.