Why is ゆうめい considered a な adj even though it ends with い? It is just one of the ones in the exception pile?

"i adjectives" are native Japanese adjectives originally going back to old Japanese. Some grammarians call them "true adjectives". There is only a limited number of them. "na adjectives" are made up of words borrowed from Chinese. ゆうめい happens to be one of those, that's why it's not an "i adjective". A "na adjective" is originally a noun, that's being used like an adjective, with "na" added as a way to mark that it's functioning as an adjective.

Etymologically, the question is, why does this word end with an "i"? If it's an old Japanese root that has had "i" added to the end so it can be conjugated accordign to the "i adjective rules", it's an "i adjective". If it ends in "i" just because it's made up from a Chinese word that just happens to end with an "i" with no connection to Japanese grammar, it'll be a "na adjective".

You can tell by looking at it written in kanji

Yuumei is spelled 有名. The "i" is just part of the on-reading for the kanji 名 ( mei) inherited from Chinese, it doesn't mean anything in Japanese grammar.

But if you look at an "i-adjective" like atarashii, it's spelled

新しい - the "i" has been added to it as okurigana ( hiragana trailing after the kanji), because it's been purposefully put there for Japanese grammar reasons, not because it's coincidentally there as part of some borrowed Chinese reading.

Incidentally, if anyone is interested in learning more about this kind of thing, I can recommend a very old grammar book "A Handbook of Colloquial Japanese" by Basil Hall Chamberlain, published in 1887 and long since out of copyright and can be found on the net. Obviously the language has changed a bit since then, so his examples have to be taken with a grain of salt... but he goes into some detail on the the history of the language and the way it ended up the way it is now. You realise that a lot of things that just seem like random and confusing exceptions to rules turn out to have some historical basis to them. I would recommend it for intermediate learners.