Well, you may call me a Philistine if you want to, but I’m a native Portuguese speaker, and to me saudade is just the sadness, which can range from really sad to sweetish, you feel because you can no longer see someone or do something that you like. So maybe there is no single English word for this, but there surely is a single English word for ter saudade de (have/feel saudade for/of) or estar cheio de saudade de (be filled with saudade for/of), which are the most common usage of saudade. The word is miss in the sense of (Oxford Learner’s Dictionaries):
8 [transitive] to feel sad because you can no longer see someone or do something that you like
The Houaiss definition given in the question is quite mind twisting, but the examples that go with it (in my 2003 Portuguese edition) are quite straightforward. I now translate them (I added eu tenho, ‘I have’, to the Portuguese original, which I give in brackets):
- [I have] saudade for a friend who is now living far way → I miss a friend who…(s. de uma amiga que hoje vive distante)
- [I have] saudade for a dead relative → I miss a dead relative (s. de um parente falecido)
[I have] saudades for the Algarve → I miss the Algarve (saudades do Algarve)
[I have] saudade for eating papaya → I miss eating papaya (s. de comer papaia)
[I have] saudade for the beach → I miss the beach (s. da praia)
[I have] saudades for the homeland → I miss the homeland (saudades da pátria)
[I have] saudade for the good times → I miss the good times (s. dos bons tempos)
As for the Rio 2016 Olympics closing ceremony, here’s a headline and my word-for-word and idiomatic translations:
Olimpíada Rio-2016 vai deixar saudade nos cariocas e turistas
Rio 2016 Olympics will leave saudade in cariocas and tourists
Rio 2016 Olympics will be fondly missed by cariocas and tourists.
So that’s all the uniqueness of saudade boils down to: Portuguese speakers use an all-purpose verb, ter (to have), and a special-purpose noun; English may lack an equivalent special-purpose noun, but has a special-purpose verb to express the whole idea. But please, please, don’t tell anyone. There’s an industry out there waxing lyrical about the uniqueness of saudade, and my fellow native Portuguese speakers will crucify me if they get to know I’ve let the secret out.