Doesn't the Septim bloodline have some magical powers to keep Daedra at bay? Nobles and such would want a figurehead around for practical as well as symbolic reasons. This ended in Oblivion, leading to their decline.
Come Skyrim and Imperial nobles will hire Dark Brotherhood to assassinate an Emperor they dislike.
They were in decline during the second era due to Molag Bal invading Nirn, but that was before Tiber I believe. They were prominent before the second era due to them obliterating the elves of Cyroriil and establishing the Empire in the first place.