So, first of all Mint Linux, and Ubuntu are both derived from Debian stable, which in case you don’t know, is the Official GNU Linux. Debian is the release that Linux Torvolds (the father of Linux) tests. All distributions Manjaro, Gentoo, must derive their base code from Debian, or they cannot use the Linux name. Its a requirement to associate the Linux name.
It appears, that you didn’t change the flag to executable under permissions, and it won’t execute if it is not marked executable. Unlike Windows, there are many file types that can be marked executable and will execute include bash scripts and python programs as well as others.
It does not matter what desktop you use, they all work if the proper dev libs are installed. But I don’t suggest you start fooling with adding or removing GUI libs if you aren’t sure. That is a sure way to end up with a command line only system.system.
Linux Mint is also based on Debian and directly derived from Ubuntu. If if runs on Ubuntu, it will run on Mint, and I know it works on Mint 21. You don’t need to wipe out Mint for Ubuntu, Mint will execute Lightburn just fine.
When you install Lightburn on Linux, you will see an entry in your applications menu under the Graphics tab that will execute Lightburn, and yes it works and installs on Manjaro just fine.
Unfortunately, there seems to be little documentation that is Linux specific after the installation, but it does install and work properly. Even the cross hairs that shows where the laser is positioned within the current project works but depending on your screen geometry the cross hair may not show without zooming in on the pointer. Lightburn does seem to have problems with GTK3 on Linux when it comes to determining proper screen metrics.
Linux is much more system friendly than Windows if you understand how it works. The current version of LightBurn works fine on every version of Linux that is current, including Manjaro, which really shows off how fast lightburn can run. Linux is not version dependent, and in fact, the Windows version of Lightburn will properly install and run on both Linux Mint 21 and the latest Manjaro providing you have Wine and Mono installed prior to LightBurn. I do not recommend installing the Windows version of LightBurn on Linux unless you know how to manually map the /dev/ttyUSB devices to Windows COM ports.
Finally, you need to be aware that if you turn off your laser engraver using GRBL, you should also unplug the USB cable to the engraver afterwards. Otherwise the engraver is still seen as an active COM port and may not initialize properly when turned on. This generally shows up as a different ttyUSBx device number. Otherwise LightBurn works without any issues at all on Linux. Debian 11, Ubuntu 22.04, Mint 21, and Manjaro all work properly.