Wohoo!
Do you want to tell why this has been an obstacle, given you got a Linux version running all fine and Steam supports Linux?
The problem was setting the appropriate file permissions for the Linux executables. I don't have a Linux machine, but I do have a tiny command line application for Windows that can set the executable bit for the appropriate files and pack the build into a tar.gz archive, thus preserving these permissions. But while Itch.io supports uploading tar.gz files, Steam doesn't; it's either direct upload or zip files.
What changed things is that recently I was given a macbook at my job, and because macOS is built on Unix, I was able to set all correct permissions there and upload my files directly from that laptop. It was a bit complicated to set up at first, but now it's running like a charm.
I talked about this with fellow Linuxers. My gist would be: Do ask, we're mostly nice people. :) But someone came up with a technical solution as well (we all know, social stuff can be hard ;) ):
https://www.gamingonlinux.com/2021/03/plus-x-is-a-simple-tool-to-help-developers...