The Local tab is also, as far as I've found, the only place where you can see how well a protracted battle is going in terms of raw numbers.
It's also easy to RMB center on tactical targets of importance, as well as quickly navigate your own assets.
If I want to tell all of my Fusion Bomber Wings to prioritise a target I can do that in the Local tab with two clicks. Can't do that from the Fleets tab.
All of the above actions can be done from the Planet map, sure, but far less easily as quite often you're looking at a giant melange of icons.
I do use the Fleet tab quite a lot; it's just as handy as the Local tab for moving assets from planet to planet. It's when you get into battles that I find the Fleets tab doesn't provide the same level of help, or offer the same amounf of information, that the Local tab does. Maybe there is a way to combine the two, or augment the FLeets table to take over some of the battle handiness of the Local tab.