Here is a Reply from Fr. John Matusiak on this very question.
There is a tradition, especially among the Great Russians, where the Royal Doors are left closed throughout the Liturgy. When there is need -- for example, for the Little and Great Entrances, reading of the Gospel, distribution of the Holy Gifts, etc. -- the doors are opened. At all other times -- specifically when there is no liturgical action which would require opened doors -- they are closed. At certain points, such as during the Litany which follows the Cherubic Hymn or during the Communion of the clergy, the curtain is also drawn shut.
Rubrics for the opening and closing of the Royal Doors is no where more complex, I believe, than in the celebration of the Vigil -- Vespers and Matins -- where it is quite clear that the doors are opened when there is a need for someone to exit the altar while they remain closed when no obvious reason exists.
Further, within the Russian tradition, the possibility of celebrating the Liturgy with the Royal Doors opened, either for greater portions of the Divine Liturgy or for the majority of the Liturgy, is often granted to clergy as a mark of distinction for outstanding service or in recognition of an individual priest's ministry.
What I have described is not necessarily common among Orthodox Christians of other traditions, such as among the Greeks or Antiochians. In some cases, in place of the Royal Doors as we know them, a large panel, often with an icon of Christ portrayed as the "Hierarch of Hierarchs," stands in their place. In such cases the icon, which generally completely covers the entire opening of the doorway, is placed on a track by which it might be slid to the side during services. I have even visited churches in the Middle East, some of which are hundreds of years old, in which one finds an iconostasis but no doors whatsoever, even though there are openings.
What is unusual, at least in my uneducated opinion on Byzantine Catholicism, is that a practice which is common among Russians, but surely not common among Byzantine Catholics -- many of whose ancestors come from the Carpathian mountains which span what is presently southeastern Poland, far western Ukraine, eastern Slovakia, and portions of Hungary -- would be introduced among people who traditionally had not known the custom. In fact, in many Byzantine Catholic churches in these regions, the iconostasis as we know it does not exist. Many Byzantine churches in the US did not even have iconostases until recent times, and especially after Vatican II urged the Eastern Rites to rediscover and preserve their heritage in its fullness rather than to continue on a course of latinization.
Perhaps in earlier times, before the Unions of Brest-Litovsk and Uzhorod, every church in the Carpathian region did in fact have an iconostasis and did in fact follow the Russian practice of closed doors, but this would be very difficult to confirm for a variety of reasons, not the least of which is that since this period is at least 400 years in the past, no one personally remembers it.
So, with regard to the specific scenario you describe, it is correct, on the one hand, to say that celebrating the Liturgy with the doors closed for a greater portion of the service is indeed "a tradition," but it is not necessarily accurate to imply that it is a tradition which is indigenous to the regions in which Byzantine Rite Catholicism was introduced. The real question, then, revolves around why this is being introduced: as a return to full Byzantine Rite tradition [as opposed to the Byzantine Rite tradition of the Russian Typikon]? as a restoration of the full Byzantine Rite as observed in Ruthenia [where many churches did not have iconostases and did not have to contempate whether or not doors would remain opened or closed]? or as a matter of taste and preference on the part of the pastor?
With regard to seeing what the priest does, I do not mean to be glib, but as a priest I would have to say that there is very little to see. Most of the time the priest stands at the altar in simple prayer. Of course, there are certain actions, such as the waving of the aer over the Gifts during the Creed, the making of the Sign of the Cross over the Gifts during the Epiklesis, the elevation and fractioning of the Lamb, etc., which one may not be able to see when the doors are closed, but in general even when the doors are opened there is little to see. We might be reminded of the words of Saint John Chrysostom who said "Christ will not appear until the priest disappears." The goal of our liturgical worship is to look beyond the priest, beyond his individual physical or psychological characteristics, and beyond his unique personality, and to come face to face with the Living God. If we do not encounter Him in our worship, even the celebration of the Liturgy facing the people, as in the contemporary Roman Rite, would only serve to enlighten us as to the priest's actions, and nothing more. Watching the priest's every action can, in fact, become an end in itself and an obstacle to "keeping watch" to the Lord's every action.
Hope this helps!