That's a great idea, thanks!
The more complex save files often get over a megabyte in size and trying to paste the save data to the browser directly would likely be pretty slow. When I last tried playing around with the saves on site I could freeze the browser trying to access the saves, but that was years ago and directly from the localstorage tab in the inspector. The save/load script I made (possibly the same you're talking about) accesses the localstorage from javascript and doesn't seem to lag much at all, so making a function that could load save data from the client shouldn't be impossible. I'll try if I can make a js script that could load the contents of a text file into the localstorage, that should make it possible to load doodles from the client.
The client technically can produce save files that the site can read, but the functionality is not linked to any button, so I'll have to add some keybind to save the doodles in that format once I get the javascript working.
edit: I got the javascript mostly working but it's late so I'll continue later.