Business: how long would this pile of vague tasks that are not fully defined, and whose requirements we don't even know, take to finish?

Forever

It will take as long as researching the actual use cases and requirments does plus whatever time the actual implementation takes plus whatever time bug hunting and change requests take.

For such tasks i normally give the honest estimate of "it is done when it is done".

developer: at least one month longer than you're willing/able to wait/pay

Time = { x | x < ∞ }