I very much doubt you are incompetent; some things just seem not to click with people, especially if they have a teacher that does not have a style that works for them. Alternatively when they say you should be spending ?? hundred hours on a module outside class they may well mean it.
I can not find out what you are studying, presumably some kind of maths involved subject but that could be anything from engineering to physics to maths to electrical engineering to computing. If I want to relate it back to something then it might help
"Anybody else feel like talking about logic in such a rigid way is just not natural?"
I find the American approach to teaching maths to be truly silly, the naming/application approach being a big part of that. Even when talking about maths problems if someone pipes up with we should look to discrete maths then it sounds a bit like
https://xkcd.com/451/ to me. Fortunately nobody I deal with (so mainly those in Europe) speaks like that.
As for the matter of logic being discussed like this. Maths beyond basic arithmetic has always felt like abstract game rules to me anyway, even stuff I can clearly demonstrate in the real world.
"Why do I feel like I get the concepts and then I just fail."
Have you got a marked exam back to look at to see where you went wrong? Depending upon the module (and a lot of maths sees people get things all right where more waffly things are harder to do that in) then if they are doing that grading on a curve crap then it could well be only a few things that send you down adjusted grade percentage.