I always thought that Hawking was confused on his philosophical alliance. For one thing, Stephen has stated many times that he believes in the existence of the laws of physics, which is quite a metaphysical position. The positivists would not be fond of that kind of view. Hawking would probably fit closer to the philosophical view that has replaced logical positivism, which is logical empiricism (don't laugh, this is what has replaced it).
Let me state the view of logical empiricism by one of the great empiricists of our time, the late Wesley C. Salmon:
"The fundamental tenet of logical empiricism is that the warrant for all scientific knowledge rests upon empirical evidence in conjuction with logic, where logic is taken to include induction or confirmation, as well as mathematics and formal logic... The roots of logical empiricism are so intimately entwined with those of logical positivism that the two are often mistakenly identified with one another." ('Logical Empiricism', Wesley C. Salmon, A Companion to the Philosophy of Science, Ed. W.H. Newton-Smith, 2000).
I think this view fits more closely with Hawking and many other empirical scientists worldwide. Although, because of Hawking's metaphysical beliefs on the laws of physics (which he calls 'God') and therefore metaphysical 'cause', he might be somewhat outside the logical empirical camp that seems to look upon these issues in a Humean fashion (i.e., David Hume) who criticized any metaphysical notion of law and cause. Getting more specific is tough without knowing more about Hawkings' views as well the empiricist reaction and tolerance for such Hawkingean views.
Why did positivism bite the dust in the 60's?
Oh gosh. A number of reasons. Quantum physics was probably a significant factor in that I'm not aware of one of the quantum guys going into the positivist camp. Einstein left the camp with his later views on general relativity. The drawomg effects of logical empiricism pulled many away from positivism. When Carnap left the camp in the 50's for logical empiricism, the movement was all but dead in philosophy (although I think it still lingered in psychology into the 60's). The most important reasons had to do with philosophical weaknesses of the movement. They tended to deny scientific explanation, rather they saw the role of science as to describe and predict as well as to categorize our knowledge. Also, they saw meaning as something that comes only from verification of experiment/observation, so if a theory has unobservables, they denyed those unobservables as true or false (which I'm sure went over well with the quantum guys), and many other reasons. What happened in philosophy in the 40's and 50's is that positivists came under heavy philosophical attacks by people like Reichenbach, Popper, et al.). The movement was unsuccessful in its attempts to create a reduction to scientific statements to either logic (i.e., logicism failed at this time), and they couldn't reduce science to operationalism (i.e., the view that all scientific knowledge can be reduced to scientific observables). With these and other collapses, the movement spurted out.