Antidoron does a particular piece of work, and it is easiest to see by asking what an Orthodox parish does with a visitor. Communion is restricted to Orthodox Christians who have prepared by fasting and confession, so at any given liturgy a large part of the congregation may not be receiving: those who have not prepared, those under a penance, catechumens, and anyone present from another church or from none. Sending all of them away with nothing would make the closure feel like an expulsion. The blessed bread is what is given instead, to everyone in the building, with no questions asked at the point of giving. The table stays closed and the hospitality stays real, and neither is asked to pretend about the other.
The spread of the custom is usually connected with the long decline of frequent communion in the Byzantine world, which left most of a congregation at most liturgies with nothing to receive; the antidoron filled a gap that infrequent communion had opened. The West had an equivalent for the same reason. Blessed bread, the eulogia, was distributed after Mass to those who had not communicated, was widespread through medieval Europe, and survived in French parishes as the pain benit well into modern times before falling out of ordinary use.
The two customs have had different fates. The western one faded as frequent communion returned in the twentieth century; the eastern one did not, and instead acquired a second job as the ordinary way an Orthodox parish can receive a guest of another confession without either church implying that a division has been settled.