All I want is to be able to define a search, and have gmail separate those into a new tab, with a badge that tells me how many unread emails are there. That's all I want and my productivity would increase drastically.

They almost have it with multiple inboxes, but the layout is really shitty and I have to scroll around my gmail window to see the various inboxes. Why can't they just add this as another tab? It's infuriating that such a simple and useful feature doesn't exist yet. This is why having a monoculture for things like email is a bad idea, they don't really care.