I can get a looping GIF to work using the /_/theme/file.gif method. Have you verified that the GIF does indeed loop? Sometimes when you edit a GIF, the image editor stops it looping - Mac OS’s Preview app does that.
https://anvil.works/build#clone:OFTJIDUY6IVIG2KJ=4VHNLAWLFCDRHADPX7L6R6FQ
(Yes, I know it’s not obvious from that that the GIF is looping, but I thought it was pretty… and if you clone it and watch it for an hour you’ll realise it must be looping!)