I can find quite a few questions on Keyboard Maestro asking how to do this. Most solutions seem to involve simulating selecting menu items and keystrokes. In my experience simulating key presses runs into problems, especially when trying to get the length of pauses right.
Is there a way to save a PDF to my Desktop with a single keyboard shortcut without having to simulate selecting menu items and pressing keystrokes? Maybe running a script?
There's a possibility that AppleScript may have access to the content of the web page in the open tab in Safari and can access that content and save it without using the Safari menus. I'm not able to help with that, but I am having trouble understanding why you don't like simulating keystrokes.
You can print from applescript, saving to PDF is a bit different. I ran across this, which might help. But I agree with @Airy .. scripting menu / button pressing is extremely reliable and I wouldn't hesitate doing that. Unless you want to surreptitiously capture somebody's web browsing, then I'd just use screen shots.
It would be difficult to be more efficient than this approach, but if it matters it's worth noting there are differences in the PDFs generated using File > Share versus Print to PDF.
File > Share: The PDF is more likely to retain more of the original site formatting, things like background colors. However,
Print to PDF will include in the PDF page footers that include the URL of the site and time stamps from when the PDF was generated. The Share method has this information, but it's only accessible in the Finder when viewing the Get Info for the the PDF.
A test of PDFs of a web page made using the two methods will demonstrate the differences for comparison, if the distinctions matter.