The Issue AirDrop is as old as the iPad … seems to date back 2015 !!!
Apple should direct all AirDrop received file in a specific folder.
as Windows and Linux does.
then user may / can move the file to appropriate folder
In Feb 2022 with my iPad Pro 11” iOS 15.3 I am having same issue.
I sent a .PDF file from My iPhone to my iPad , all status ok but can’t see the file on my iPad.
iPad Pro, iPadOS 15