The celebration of holy michael day by C&S church is not because other churches (orthodox churches) are celebrating it.

Although the origin of St. Michaelmas day popularly known as Holy Michael day.

The early Church fathers are the apostles who took over from the original followers of Jesus Christ, the are the one who formed Church doctrines, creed and some church teachings.

It was in one of these early Church councils where all the stake holders of the early church (the Apostles) were having conference that they looked into the church in the wilderness which was during the time of Moses who led Israelites out of Egypt.
Exodus 23:20
"Behold, I send an Angel before thee, to keep thee in the way, and to bring thee into the place which I have prepared. 21 Beware of him, and obey his voice, provoke him not; for he will not pardon your transgressions: for my name is in him."

God gave Angel Michael to the Israelites to guide, protect and lead the Israelites through the wilderness to the promise land.
Exodus 33:2
"And I will send an angel before thee; and I will drive out the Canaanite, the Amorite, and the Hittite, and the Perizzite, the Hivite, and the Jebusite:"

So, the church council, having the understanding that Christian Church is the Spiritual Jew as seen in Romans 2:28
"For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh:
29 But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God."