Is it correct that neither “worthy” or “able”, are not so valid translations, as acclamation of a new Emperor or Patriarch,like the Greek word Aξιος ?

As Dimitrios Michmizos says, “worthy” is the best translation; “worthy” is not about one’s worth, “valuable”, but about one’s merit. It certainly isn’t idiomatic in English as an acclamation, though.

Konstantinos Konstantinides points to the added gloss “deserved”; and while it is not a common expression in English, you will occasionally see the exclamation “Well deserved!”

Seth’s Blog: “Well deserved”

“Congratulations” is fine for winning the lottery, but “well deserved” is reserved for people who put in the effort and the time and took the risk to get somewhere.