“This is the best of [Ø] X” means “The best of all X”; the best of the X” means “The best of [some specific subset] of X”.
So “the best of times” here means “the best of all conceivable times”. If Dickens had said “It was the best of the times” it would imply some such qualification as “It was the best of the times anyone living had experienced.”
This phrase needs to conclude on a plural rather than a collective. “Men”, “cows”, “chickens”, “suitcases” all work, but “The best of humankind”, “the best of cattle”, “the best of poultry”, “the best of luggage” are all very odd. These collectives semantically reject partitioning into individual members, so there can be no “best” of them.