This is handwaved somewhat in the "Resolving problems" section, but mis-location of (non-extension) DLL files is often a problem when installing PHP extensions on Windows.
Many PHP extensions come with not only the extension DLL, but supplementary DLLs that are required by that extension. (For example, php_luasandbox.dll comes with lua5.1.dll, the lua interpreter it sandboxes.) Those other DLLs should go into the same directory as the php.exe binary, NOT the extension directory.
So, if php_luasandbox.dll is installed at C:\PHP8.1\ext\php_luasandbox.dll, the interpreter would be located at C:\PHP8.1\lua5.1.dll. That allows the PHP binary C:\PHP8.1\php.exe to find those additional DLLs when required.