"I guess "Желаю тебе" is omitted in the phrase "Всего́ до́брого"" - You're absolutely right. In sentences that begin with "I wish you..." in the Russian language is used genitive. "Всего" is the genitive case of the pronoun "всё". In expression "Всего доброго" the word "добрый" is used to mean "хороший", "отличный" ("good", "excellent"). "Всего доброго" = "I wish you all the good" ~ "I wish you all the best". It is the formulа of the farewell, the farewell wishes.

"Всего" is genitive case of "Всё" "Я желаю тебе всего хорошего/доброго/наилучшего" can be translated as: "I wish you all the best" желаю -> кому? -> dative -> тебе желаю -> чего? -> genitive -> всего