Installed the game. Clicked "Play" on GOG Galaxy. Got the launch window and then when clicking "Start" nothing happens.

Here's what I have tried
1) changing the settings in the launch window
2) installing various versions of the Visual C++ Redistributable Packages
3) running the launcher on various compatibility modes
4) running the game without the launcher window, i.e. opening RebelGalaxyGOG.exe directly
5) used the verify / repair function and update function on GOG Galaxy

Running on Windows 10. I have admin status. And I know that in 2019 I played the game on the very same laptop I am using right now.

Open to any suggestions.

Which ones exactly?
I think you need the 32-bit version of Visual C++ Redistributable for Visual Studio 2012. You may need the 64 bit version as well, even though I'm not sure about this.
If both 32 and 64 bit packages are already installed, you may want to try repairing them using the installers taken from the link provided.

Right now I have the following installed:

Microsoft Visual C++ 2005 Redistributable (8.0.61187)
Microsoft Visual C++ 2005 Redistributable (x64) (8.0.61186)
Microsoft Visual C++ 2008 Redistributable (x64) (9.0.30729.7523)
Microsoft Visual C++ 2008 Redistributable (x86) (9.0.30729.7523)
Microsoft Visual C++ 2010 Redistributable (x64) (10.0.40219.473)
Microsoft Visual C++ 2010 Redistributable (x86) (10.0.40219.473)
Microsoft Visual C++ 2012 Redistributable (x64) (11.0.50727)
Microsoft Visual C++ 2012 Redistributable (x64) (11.0.51106)
Microsoft Visual C++ 2012 Redistributable (x64) (11.0.60610)
Microsoft Visual C++ 2012 Redistributable (x64) (11.0.61030)
Microsoft Visual C++ 2012 Redistributable (x86) (11.0.50727)
Microsoft Visual C++ 2012 Redistributable (x86) (11.0.51106)
Microsoft Visual C++ 2012 Redistributable (x86) (11.0.60610)
Microsoft Visual C++ 2012 Redistributable (x86) (11.0.61030)
Microsoft Visual C++ 2012 Additional Runtime (x64) (11.0.61135)
Microsoft Visual C++ 2012 Minimum Runtime (x64) (11.0.61135)
Microsoft Visual C++ 2012 Additional Runtime (x86) (11.0.61135)
Microsoft Visual C++ 2012 Minimum Runtime (x86) (11.0.61135)
Microsoft Visual C++ 2013 Redistributable (x64) (12.0.21005)
Microsoft Visual C++ 2013 Redistributable (x86) (12.0.21005)
Microsoft Visual C++ 2013 Additional Runtime (x64) (12.0.40664)
Microsoft Visual C++ 2013 Minimum Runtime (x64) (12.0.40664)
Microsoft Visual C++ 2013 Additional Runtime (x86) (12.0.40664)
Microsoft Visual C++ 2013 Minimum Runtime (x86) (12.0.40664)
Microsoft Visual C++ 2015 Redistributable (x64) (14.0.24212)
Microsoft Visual C++ 2015 Redistributable (x86) (14.0.24212)

Everything appears to be ok with this requirements, anyway you may want to try repairing the packages as I suggested in the previous post. And don't forget to reboot the system afterwards, in order to make the changes effective.

Did the repair thing, with the installers that offered that option.
Did restarts when installing (and deinstalling and reinstalling) the redistributables.

Even got an additional copy of the game from Steam, which also failed to launch.

Does the Steam copy prompt any error message whenever you launch the game?

Which render do you select in the launcher? At this point I'm suspecting that the issue could be related to DirectX.

I have tried all three options: Direct3D 11, Direct3D 9 and OpenGL.
I have tried all of them on both fullscreen and windowed.

(and as I said in the first post - I have played the game back in 2019 on the very same laptop I am trying to play it right now)

Typically this kind of behavior can be caused by the previous uninstall of the game not cleaning out all of the mentions of the game. I would try to uninstall the game and then run something like ccleaner to clean out the registry of mentions of this game. Once done then try and reinstall the game.

Will try this next.
Although I am beginning to lose hope.

Uninstalling the game (including previous saves and settings), clearing the registry of Rebel Galaxy and re-installing the game did not help.

I "found" an older version of the GOG installer of the game, that one came with some C++ Redistributables. But that was also no good.

I am giving up on this. Just have to be content with listening the soundtrack on YouTube.

Hey there! If you just want to listen to the music it is all in its own music folder. Just use most music player programs to play the music.