Hi Daniel! In order to come to the accepted text of the Byzantine type, the Church still had to work through the multitude of variants. No two manuscripts are identical. So, in order to even arrive at a "Byzantine text type", the accepted texts had to be arrived at. That is why I am saying that they have, of course, engaged in textual criticism.
I don't know that we can speak of the accepted text, the Patriarchal text which is "the accepted text" of the Ecumenical Patriarchate did require the going through of various manuscripts in order to have a standard Greek text for all churches within the Patriarchate. But the way in which this was done (i.e. selecting one manuscript, rather than creating a composition or "Majority/Critical text") is not in accordance with modern textual criticism, and this one event does not override the fact that the majority of Orthodox Scriptures (Slavonic ect..) have not undergone this process. So though to some degree the Church has engaged with textual criticism (if defined quite loosely) it is not a Orthodox practice par excellence and certainly not the main concern of the Church when it comes to the Scriptures.
The Byzantine text type itself is not an arrived at text, it is a term used by scholars to describe what they find to be a grouping of manuscripts roughly defined by geography and having readings which are not found in the Alexandrian text type and not having those found in the Western and Caesarean text types. it is not a product of textual criticism in itself it is rather used for textual criticism, the Church did not go through manuscripts and create the Byzantine text type it is rather a scholarly grouping which corresponds to the texts the Church has received.