ubiquitin,
“In addition to the 4 Rosh Hashanas you reference In September, I wish hatzlacha to the new year in school. I and they consider September to be the start of a new school year Ialways wish them success on the new year in September, or Elul as the case may be. Is this allowed?”
You are ignoring context to quibble with Little Froggie over semantics. The school year legitimately and appropriately affects us, and is clearly an arbitrary date set for convenience by school systems. The medical school year is just another type of school year. Fiscal and tax years affect us too, and are purely secular, though it’s a bit strange to wish people happy fiscal new years. But how should the non-Jewish religious observance of New Years Day affect us as Jews?
You may ask why then did LF have to bring in the other new years listed in the mishna, but I believe he did so because all of those new years are religious in nature, as is New Years Day.