The goal of it is to essentially replace the pinned tab (at position 1) with the URL I am currently viewing.
The issue is that Set Safari URL to action fails to do it. It doesn't actually replace the pinned tab's URL but opens a new tab with the URL (not pinned).
Is there a way I can replace URL of pinned tab? Perhaps with AppleScript or JXA?
If you're still using an older version of macOS, I don't know if something might have changed between that and Mojave 10.14.5, but the Set Safari URL action works fine for me with Safari 12.1.1:
But if that doesn't work for you for whatever reason, you can try this AppleScript and see if it does the trick where the KM actions didn't:
tell application "Safari"
tell front window
set FrontmostTab to current tab
set URL of tab 1 to URL of FrontmostTab
close FrontmostTab
end tell
end tell
If you're also on 10.14.5 and neither of those methods work for you, I'm afraid I don't know what else to tell you. There is nothing unique about pinned tabs in terms of AppleScript that I can see, and both of those methods work fine on my own 10.14.5 install. Just to make sure, are you using the same version of Safari too (12.1.1)?