The photo above is of an Orthodox chalice veil. This is used to cover the gifts, but the actual coverings placed over the vessels in the Great Entrance are usually shaped like a cross, so that the bars of the cross hang down at four sides. The Great Entrance takes place after the gifts have been prepared. The clergy carrying the gifts emerge from the door to the right of the Holy Table (their right) and come into the church singing. They then walk round to the centre of the Iconastasis and enter the Holy Doors, placing the gifts on the Altar or Holy Table. The gifts, even now, represent Christ, that is, Christ before the Crucifixion. The coverings represent His clothes which are taken from Him before He is laid on the Cross.
When the gifts are brought into the body of the church, the congregation shows reverence. Even before the consecration these things are taken to be holy and as representing the Saviour.
In the West, we hardly show any reverence towards the unconsecrated elements, but in the Old Rite, the veils are important, and even before the Offertory itself, the gifts are veiled and treated with respect. We need to recognise that as soon as the bread and wine (and water) are placed on the credence table - and when the priest's host is brought in procession by the priest at the beginning of the Mass - they are, to some extent, "holy" and deserving of respect; they have already been set aside for God and therefore belong to Him. In the Ordinary Rite, we need to consider how to express something of this. We might consider how the gifts are brought up in procession. At the same time, there is need to consider the Sacred Vessels which really should not be used in the offertory procession. Sometimes the elements are consecrated in unworthy vessels (even straw baskets!). The Vatican has made it clear - more than once - that only "unbreakable", worthy vessels are to be used. However beautifully made, ceramic chalices and patens are forbidden - because they break. At the same time, there is much to be said for retaining the use of gold-inlaid vessels because this refers back to the Holy of Holies (the "Golden Room") and the Ark of the Covenant.
By reverencing the offertory gifts in some way (for example by incensing them) we are pointing to the fact that everything offered to God, and accepted by Him, is holy and precious. This includes ourselves. Showing reverence in the Liturgy - to each other as well as to the gifts - may help us to remember that we owe each other respect as sons and daughters of God and that we owe respect to the whole of God's Creation. Ritual is not empty unless we misuse it, abuse it, misread it or misapply it. Those who want to celebrate the Liturgy in a penny-pinching or demeaning way do disservice to themselves as well as to their Creator.
You see, veiling has much to commend it, since it speaks to us of holiness and Mystery.