I'm having the same problem on Linux Mint, looks like it installed successfully, but no icon and I can't find the folder that it's installed in. A search for Vintagestory only brings up an empty folder in Pictures. That command (flatpak run at.vintagestory.VintageStory) does cause it to run, but is that running the copy on my computer or running direct from the flatpak somehow? I'm also new to Mint, so a little out of my depth, but the instructions on the wiki are not lining up with my experience.
I'm just looking to get an icon to launch from if anybody can help, thanks.