Okay. Here's a video of the import process in Godot 3. There's a plugin in the Godot 4 asset library that has the same features (requires the .JSON metadata paired with the .PNG file you'd like to us) called "Aseprite Importer".
This Youtube Video shows how it worked with that addon
However, after seeing your comment I attempted to repeat the import process and it was non-responsive. So the problem is probably with the tools and not how you're using them. I'll get on figuring it out - I may end up having to request permission to alter the plugin and publish it, or possibly switch up the format so to use one of the newer importers that directly take .aseprite files. I imagine whatever the fix is, it will take some time.
In the meantime, have you been able to open the sample project? everything is already imported to a Godot 4 project in a zip file (Gumdrop_Godot_4.zip). If you import that project there's an already functioning scene which includes the sprite with weapon and flash layers all animations, and an animation tree to manage everything. Hopefully that holds you over while I develop a solution.
Finally, if you're in a hurry (game jam for example) its likely someone in the Godot community could find a workaround faster than I can publish a full update. The most important thing is to use the .JSON files which contain the Aseprite spritesheet metadata, and not the raw .aseprite file which isn't intended for direct import.
Thanks again for bringing this to my attention, and thanks for your interest in my animations. I'll drop a reply to your reddit post whenever I'm ready to publish a fix.