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This is not correct (and is the same mistake he is making).
If you look at the beggining of Rosh Hashana the Gemara goes through several possible rayah that We start the year for Jewish Kings from Nissan , and for Goyish kings from Tishrei you will see this clearly.
for example The Gemara says how do we know Jewish Kings are counted from Nissan?
Answers Hekish to Yetzias Mitzrayim.
Asks How do we know Yetzias Mitzrayi is from Nissan?
Answers It says Aron died in the 5th month of the fortieth year ie Av then says Moshe spoke in the 11th month of the Fortieth year ie Shevat So you see within a year Av is before Teves which means new year cant be Tishrei.
CLEARLY the fact that the Torah calls Av the 5th month doesnt tell us anything about when the year starts. In fact the Gemara entertains the possibility that the story of Shevat (ie the 11th month) precedes the story in Av (the 5th month) all in the 40th year!
In other words if the year started in Tishrei (as it does for most things) The first month is “chodesh Hashvii” the second month is “chodesh hashmini” etc etc
these are the NAMES of the month .
The Torahs’s name for Av is “Chodesh hachamishi” whether it is the 5 th month of the year or the 11th,. The NAME of the month is Chodesh Hachamishi.
Many over the years have discussed why we no longer use these names/ and not all agree with the Ramban. but to actively change the number system doesn’t seem right