Others can probably provide more reasons why the Hebrew calendar is unbiblical or inaccurate but here are a few.
The Hebrew calendar is quite different from what is defined in the Torah.
Recall that the first month of the year took place at the Exodus.
[Exo 12:1-2 NKJV] 1 Now the LORD spoke to Moses and Aaron in the land of Egypt, saying, 2 "This month [shall be] your beginning of months; it [shall be] the first month of the year to you.
This is the month of Abib, which the Jews have re-arranged to be the seventh month in the Hebrew calendar.
Tishrei is the seventh month as commanded in the Torah, and the first day of the seventh month is Yom Teruah (Day of Shouting or Trumpets). Yom Teruah is the first of the three Fall moedim.
[Lev 23:23-25 NKJV] 23 Then the LORD spoke to Moses, saying, 24 "Speak to the children of Israel, saying: 'In the seventh month, on the first [day] of the month, you shall have a sabbath-[rest], a memorial of blowing of trumpets, a holy convocation. 25 'You shall do no customary work [on it]; and you shall offer an offering made by fire to the LORD.' "
In the Hebrew calendar, Tishrei has been placed as the first month of the year. Furthermore, Yom Teruah was changed to Rosh Hashanah (Head of Year) as the new year day.
The Hebrew calendar also contains leap years to keep sync with the solar year which Scripture never describes. This was likely implemented during the Babylonian captivity by the influence from Babylon where the Hebrew calendar eventually adapted the Akkadian names for the months.
The current Hebrew year count, known as Anno Mundi (AM), relies heavily on genealogies since Adam and there is insufficient resolution in the genealogies to come to an accurate count. There is also the problem of the 'missing years' where Jewish Rabbis and academic scholars disagree on the year the first Temple was destroyed.
I think the only accurate method we have to determine the current year count is prophecy.