The two systems measure different things: how well the war is going and how tired the nation is of war.
You can be having a late surge with new tech and a reserve fleet, be half way to the enemy capital and then have the masses pull the plug on your war because they were exhausted 2 years ago.
N.b They don’t measure particularly accurately.
I guess my point is, I don't see the point of the first measure.
How well the war is going should be part of how tired the nation is of war, absolutely, and different empire types should experience it differently. But if that late surge doesn't convince the public to keep fighting, then that's that. Or if it's enough to renew their spirits and get them back in the fight, great. Either way, though, I don't see what adding war score as an separate number adds other than confusion.