How exactly can I enable the controller or change the controls? There appears to be no controller/controls related options in the game?

I have no idea. What controller are you trying to use? I am trying to get it to work with a PS4 controller but no luck.

I have a Logitech Dual Action gamepad, which is basically just a Playstation controller clone...

The Game itself seems to be really good and everything I hoped for it to be, but I can't force myself to play it with keyboard, even though I have long history of gaming with keyboard on PC, played Commander Keen series etc. with keyboard back in the early nineties.

I read in the Steam forums that apparently there is going to be a patch released to add controls remapping, but I guess I need to fire up Joy2Key to use controller for the time being.

Can you try the keyboard to button mapper in 8Bitdo website?
It helped me with other games to map a button from a controller to a key. It's in the 8bitdo support, Xmap I believe is called