One early description of the administration of the Eucharist [the Thanksgiving] is given by Justin the Martyr. He confirms the OP's understanding. He specifically states that the "presider" of the Eucharist is of the "laity". This of course won't support the later tradition of a sacerdotal priest the OP mentions.
There is then brought to the president of the brethren 1909 bread and a cup of wine mixed with water; and he taking them, gives praise and glory to the Father of the universe, through the name of the Son and of the Holy Ghost, and offers thanks at considerable length for our being counted worthy to receive these things at His hands. And when he has concluded the prayers and thanksgivings, all the people present express their assent by saying Amen. This word Amen answers in the Hebrew language to γένοιτο [so be it]. And when the president has given thanks, and all the people have expressed their assent, those who are called by us deacons give to each of those present to partake of the bread and wine mixed with water over which the thanksgiving was pronounced, and to those who are absent they carry away a portion.
Administration of the sacraments.
NOTE 1909 τῷ προεστῶτι τῶν ἀδελφῶν. This expression may quite legitimately be translated, “to that one of the brethren who was presiding.”
Basically, the idea is, as Peter and John tell us, all believers are priests (1 Peter 2:9, Revelation 1:6). If believers come to my house, I preside, given I am an elder. If we go to the house of another believer, he presides, given that he is an elder.
PS. Some incorrectly believe that the meaning of "president" refers specifically to a "priest", rather than "elder". The Greek word translated "president" is akin to "elder" being πρεσβύτερος. Martyr, however, knows the difference between two words and offices. Had Martyr wanted to tell his readers that priests only officiated at the Eucharist [Thanksgiving], he could have said so plainly.
[Re scapegoat] the first, in which the elders of your people, and the priests, having laid hands on Him and put Him to death, sent Him away as the scape [goat].
He thus returns to the Mosaic laws, and uses it to show that the things which pertain to Christ are changed from the Old to the New. The apostles taught that all believers are priests, but some are also elders. The elder from the brethren presides. It is not the "ordained in some way" priest who leads the Eucharist, but the elder of the priesthood of which all believers are part. So again, when you come to my house, I preside. When we go to your house, you preside.