I would say that "of ancient date" sounds a bit more literary, or even poetic. The phrase "of ancient times" is more common. There is also a slight difference in meaning. When you say "of ancient date," you are referring to the date when something was produced, published, created, etc. This term usually refers to something that has been preserved, and still exists (like a written work, a piece of clothing, a tool, etc.) Something "of ancient date" is something that was made a long time ago, and we still have it. When you say "of ancient times," you are simply saying that it existed or was used in ancient times. The meaning is a bit more general. For example, you can talk about "people of ancient times." This means "people who lived a long time ago." However, you can't talk about "people of ancient date." That doesn't make sense. We don't think about people as being "produced" at a certain time. Also, if a person existed in ancient times, that person does not exist now. You can also say something like, "This was a popular food of ancient times," meaning that it was popular to eat this kind of food a long time ago. But it doesn't make sense to say "a popular food of ancient date." "Food of ancient date" would most likely refer to food that has been preserved for hundreds or thousands of years, such as an ancient loaf of bread that you might see in a museum. Both "ancient date" and "ancient times" could make sense in the sentence you quoted. I think "ancient date" sounds a little more lofty and sophisticated, but some people might think it sounds pretentious. Based on the rest of the sentence, though, it sounds like the text you were reading was written in a somewhat lofty tone. A more casual way to write the same sentence would be simply, "What ancient work would still exist?"