Welcome to the Board!
I was reading that DoCmd.TransferSpreadsheet doesn't have this limitation.
I think that depends upon which version of Access you are using. If you read something that suggests otherwise, please post that link here.
I believe that the issue is that Access 2003 only has the ability to export to Excel files with a ".xls" extension, and not the new ".xlsx" extension introduced in Excel 2007. And Excel files with the ".xls" extension only allow to export up to 65,536 rows.
Here are a few options/workarounds (there are others):
1. Upgrading Microsoft Office to 2007 or newer to take advantage of this new option.
2. Exporting the file to a text file, where that text file can now be imported into Excel.