how to push a .pdf to a usb-stick from a MacBookPro?

Drag and drop it, or

Copy and paste it.

Are you saying the file will not save to the USB stick? Or that the .pdf file is not readable/viewable?

How is the USB stick formatted?

Sometimes Apple cannot access external drives when they are partitioned or formatted on other systems. This is a known issue when Windows formats a drive with exFAT. Try reformatting the USB stick using macOS.

Edit: Also try using another drive for the transfer.

The limitations of which you speak are primarily caused by Microsoft asserting that Windows New Technology File System (NTFS) is Microsoft proprietary, and they intend to sue anyone who writes to that format without paying a Royalty.

Apple thinks this is nonsense, so they do not write directly to NTFS Volumes, although MacOS can create and read such a Volume.

Microsoft does not acknowledge that there are any other File formats beyond Windows.

Apple reads all other common file formats except NTFS.

Have you tried formatting the USB-stick as ExFAT for use on Windows or MS-DOS?

Yeah, you get spoiled by Linux very quickly. I get so frustrated using other operating systems with all their limitations.