There is no 'native' way to download the old Windows 10 snipping tool back to Windows 11 unless the Windows.old directly still exists.
You may restore the old snipping tool by following this non-official article: https://www.thewindowsclub.com/how-to-restore-old-snipping-tool-in-windows-11
Best regards,
Nanta
Disclaimer: The link I sent is a non-official Microsoft source, but it seems to provide relevant information on this issue.