Given that there are 3 possibilities as brought down in the Mishna Brurah 423 or the Kaf Hachaim. However, the prevalent minhag among Ashkenazim is the one that you mention.
הוספת "ולכפרת פשע" בשנה מעוברת
אבל המנהג הרווח הוא, לפי הספרדים כל השנה, ולפי האשכנזים מתחילת השנה עד
The prevalent minhag among the Sefardim is the entire year and among
the Ashkenazim until the leap month (Adar II)
This explains the Ashkenaz minhag rationale.
Since Adar II is the last month of the year when counting by months, and Rosh Chodesh Nisan is the first month then the minhagim of Rosh Chodesh would follow that way of counting. Thus, the addition of ולכפרת פשע would end at the "end of the year" by months.
Even if you count by the change in the year number (Tishrei), by the end of Adar II, the leap month has been added and the months have been reset to synchronize with the seasons. If the Sanhedrin were to determine the leap year, the end of Adar would be the absolute last time it could have been done. As a result, we have now added the thirteenth month and the brachos for the months now revert to the normal 12. Once Rosh Chodesh Nisan has passed (without an extra month having been set to synchronize it) a leap year cannot be decreed until the following Adar.
It should be noted that the gemara mentions that Chizkiyahu decreed a second Adar at Rosh Chodesh Nisan and the Chachamim stated that he was wrong to do so. See Sanhedrin 12b.
The question would more likely be "why start saying it in Tishrei rather than the previous Nisan" if you count by Nisan as the first month.
The reason for that, would be that the earliest simanim that would indicate the the year might be a leap year would not show up until Tishrei (Rosh Hashana for the count of years). I remember seeing a reference that he chachamim would not discuss a potential leap year until Tishrei. As a result, we do not start saying ולכפרת פשע until after Tishrei. Note that since Tishrei is Rosh Hashannah, we do not have that part of musaf in Tishrei.