Pourtant introduces a contradiction, it’s equivalent to “yet” or “however.”

Your third sentence is the best example of how the word works.

J’ai pensé que le fromage bleu n’était pas bon en bouche dû à son odeur
désagréable. Pourtant quand j’en ai mangé une bouchée, j’ai trouvé ce
fromage excellent.

I had thought that blue cheese wouldn’t taste good due to its
disagreeable smell. However, when I had a bite, I found it
great.

Similarly,

Je ne comprends pas… Il y a plein de monde à la piscine, mais
pourtant le stationnement est vide.

I don’t understand – the pool is full of people, however, the
parking space is empty!

the first sentence can be used like "even"

…, even though I had made all to be accepted

the second and third can be used like "however"

…, however the parking….

…, however when I tasted…

pourtant =~ cependant

I was thinking about the contrastive/contradictory adverbs “yet” and “however”, and their equivalents in French “pourtant” and “cependant”.

“Yet” and “however” to me don’t seem quite the same. I think that they differ in this way: “however” indicates contrast/contradiction but with a stronger connotation of negativity, like a sense that the contradiction is considered unfortunate or undesirable in some way. “Yet” indicates contradiction as well, but with less of a negative connotation. The connotation may be more of a neutral tone, or even light or humorous irony.

As for the French equivalents: “cependant” and “pourtant”, this distinction may also exist, but since I am not fluent in French, I can’t say for certain. My guess is that “cependant” is also the contrastive adverb with a stronger connotation of negativity and “pourtant” has less of a negative connotation and is more neutral.

In French, ‘pourtant’, ‘cependant’ and ‘néanmoins’ have close meanings (I don’t know their differences, by the way).

I believe that ‘yet’ and ‘however’ are right translations. I think ‘yet’ is colloquial, while ‘however’ is more formal.

I would probably say ‘yet’, but write ‘however’.