How do I delete a locked file - options greyed out.

There are two lock settings:

1. The user-settable Lock selection in the Get Info panel
2. A System applied immutable lock preventing user change in the Get Info panel, and controlled from the Terminal. This will display as a grayed-out Lock selection in that panel:

sudo chflags schg ./foo.txt.   # set system lock
sudo chflags noschg ./foo.txt  # remove the system lock

In older versions of macOS, the System immutable lock would superimpose a small lock on the file icon, but not in Big Sur. Also, in the Terminal, you can see what flags have been set on a file with the following:

ls -Ol ./foo.txt

A user lock will show as uchg and a System lock as schg.

See what chflag(1) is set on your PDF, and unset it accordingly.

Try restarting in Safe Mode It will do a Repair Disk, clear cache files and only load Apple Software, extensions and Fonts. Loads Slowly - Normal. Once in run this mode 10 - 15 minutes and Test if you can remove the file via Drag and Drop to Trash. If yes - good. If not - there might be some Third Party software interfering with the process.