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Thank you so much for your post!

Hawking is not asserting he has solved the problem of first causes, he is merely describing a possible universe that does not require a beginning in time.

He gives the reason for his assertion in the lecture, excerpted below. By "that date" I presume he meant "any such date" - emphasis mine:

Thus it would require the direct intervention of God, if the universe began at that date. By contrast, the Big Bang is a beginning that is required by the dynamical laws that govern the universe. It is therefore intrinsic to the universe, and is not imposed on it from outside.

Although the laws of science seemed to predict the universe had a beginning, they also seemed to predict that they could not determine how the universe would have begun. This was obviously very unsatisfactory. So there were a number of attempts to get round the conclusion, that there was a singularity of infinite density in the past.