I suppose I don't care about the origins of the White Walkers in the same way as other people, so that definitely helps. They were expressionless, one-dimensional demons who were created for one task: destroy Westeros. They were robots with one mission and they failed. We saw how they were created by the Children of the Forest, we know about the Pact and the War for the Dawn and how they were defeated the first time. Thinking about it, having another show with essentially the same general plot as the original story would have been a waste of time. It would just be another show that built to another battle with the Army of the Dead.
At least a show about the Targaryens' history would give us a good chance to watch real people live out their lives and make hard decisions. We have a chance to watch a family tear itself apart because of how much they're hellbent on power and greed. Over the course of 300 years they go from being the most powerful family in the world, with the ability to bring super-weapons to heel, to a shell of themselves with a hideously unpopular madman for a leader. I wonder if an anthology series would be a good way to do this, with each series dropping in at various points. I don't see how you make the White Walkers that interesting.
Obviously I'm talking in massive hypothetical scenarios here, but say it manages to reach a 6th season by 2025, we could have a season per section of Fire and Blood or something. Season one would look at Aegon's conquest, season two at Aegon's reign, season three at his sons, etc. Sort of like how The Crown has set itself up to cover different eras of history. It might mean eighteen month gaps between seasons but it would be easy enough to keep a thematic throughline as we watch a family destroy itself because having everything still wasn't enough. Again, I don't see how you make a story about the White Walkers anywhere near as interesting.