She is mentioned only one time in Acts 1 - and then never again throughout the formation of the church worldwide. Not one quotation - not one action. As 'mother of the church' she has no recorded role or input. Kinda strange.
This is called an “appeal to silence” and is a logical fallacy.
There is zero scriptural record of an 'assumption'
So what? Many important things happened to the Early Church that were acknowledge by its leaders, but which weren’t mentioned in the New Testament scriptures selected by the same Church, such St. Athanasius of Alexandria. Therefore what you should focus on is the belief of the Early Church Fathers, which can be determined through their liturgical texts, which takes us to the next issue:
and the prayer mentioned in the OP was not widely use until 1,600 years later.
Here you are confusing the Rosary, which did actually predate the Battle of Loretto, but the Battle of Loretto made it popular, which is a specific Roman Catholic devotion, with the Hail Mary prayer, which is much older.
Your claim that the prayer mentioned in the OP was not widely in use until 1600 years later is demonstrably false - if you looked at the ancient Orthodox hymns known as Canons which are sung at Matins, which are mostly from the late first millenium, but in some cases even older, you will find a very common type of hymn which is built around an intercessory petition to or veneration of the Theotokos that is either the Ave Maria or a variation on it; these are called
Theotokia or in the case of those that refer to her at the Foot of the Cross,
Stavrotheotokia. You would have heard them in Greece at the Orthodox sites you visited if you had been there during Orthros or on other occasions when canons are sung. The
Theotokion is typically part of the Ninth Ode, which features the
Magnificat (the evangelical canticle
@Xeno.of.athens quoted from St. Luke) and a text based on it, which is variable.
I have cited just one example; there are several others. For example, the Russian Old Rite Orthodox used a slightly different version of this prayer than the Orthodox from the Ottoman Empire who were brought to Russia to reform the liturgy by Patriarch Nikon, and who made various changes which caused a schism, but the prayer is semantically identical, differing only slightly in terms of wording.
By the way, are you aware that Martin Luther himself prayed the Ave Maria, but not the Rosary, and not the petitionary component, and that predates 1600, and indeed he recommended the same? My friend
@MarkRohfrietsch can confirm that should there be any doubt.
Some Lutherans confused certain versions of the aforementioned
Theotokion hymn which lack an intercessory prayer (some of the
Theotokia have it and some do not) with the historical Orthodox version of the Ave Maria prayer and incorrectly assumed that our version does not ask the Theotokos to pray for us, however, the Orthodox version of the prayer, which is very old, and is also used by the Syriac Orthodox and various other Oriental Orthodox, since the first millennium, does have a request that the Theotokos pray for us.
However we must stress that you were partially correct, in that the Rosary, which is a specific rule for praying the Ave Maria in a series of decades that have a theme such as the Seven Sorrows became much more popular following the Battle of Loretto, where a Christian fleet defeated an Ottoman fleet which was preparing to facilitate a naval invasion of Western Europe, which coupled with the Ottoman control of Budapest at the time and their advance towards Vienna, would have, had they landed, likely divided the European forces, which could well have led to Vienna falling to the Turks, which would have been a dissaster and could have led to Turkocratia in much of Western Europe, and this would have been a nightmare for all Christains, whether Orthodox, Protestant or Catholic. Turkocratia was a monstrous evil but it, combined with the genocides of Timur the Lane against the Church of the East, and the previous genocides against Christians in some Islamic conquered territories, resulted in a disproportionate number of Eastern Christians receiving a crown of martyrdom.
However, that being martyred results in instant glorification does not make genocides or ongoing systematic persecution of Christians in any way desirable; rather it represents God in His perfect justice compensating Christians who confessed HIm before men and were put to death for it. So while we should pray that if the time comes, we have the courage to confess Christ and not deny Him, we should also pray for the peace of the churches of God.