I have lots of web shortcuts, they all have the 'little arrow' on... I am using Firefox portable mainly , but also use palemoon and Basilisk browsers for quick stuff that can work without needing FF.. :/

I just recently tried making a shortcut to the desktop from FF ( I am using -no-deelevate )

and it makes a 'properties' .URL file that contains only 'General, security, detail, previous versions' Tabs..

and no sign of the web address! - I can then right click this to get the option 'make shortcut' to create a .url.lnk file..
the same seems to happen with the other browsers too! :O

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also note, when I drag from the 'padlock' icon, it DOES show the 'little arrow' symbol.. until it lands on the desktop... without the arrow..