There's something to remember. The Three remain, yes, but so do
the Nine.
I believe somewhere it is said even the 3 that remain lose their power - though
Sauron had not 'sullied them' directly.
If
the One Ring's destruction destroys not only its own binding power, but the power of Sauron poured into it, surely
the nine rings would have no remaining power at all, even if their physical form remained.
The 9 whose lives they had helped prolong would be gone.
What remains is the
evil inherent in
Arda itself due to
Melkor's work, and all the evil sown by Sauron within the
peoples.
Evil such as the "
Mouth of Sauron" for example - who we know so little about, and have no idea (if I'm not mistaken) what happened to, in/after the destruction at the
Black Gate.
But that party had no ring or other 'life sustaining device' we know about either - so there is a finite direct influence the Mouth could apply.
It is the "age of man" - extended lives of all are now in even more rapid decline, and future generations are all doomed to much less than was before.
The "evil that remains" is what has been cooked into "man" himself.
My own fanfic intended to show how that evil worked its way out - all the machinations as
Elessar declined and
men vied for power, then the happenings of man after his death - ultimately leading to (in essence) what we have today. A bunch of petty squabblers never satisfied with what they have, always seeking more power and control, no rest on any hand at any time for any extehded period of time (because life-spans are now become so short) - all awaiting the "renewal of all things" when "the music is sung aright."
This was, in essence,
Tolkien's intent with the whole follow-up after Sauron's irretrievable removal.
Much wickedness left to unfold through the
ages - but as the wickedness of Sauron differed from that of Meklor, so the remaining wickedness differs from that of Sauron.