Tendency of the rate of profit to fall
I'm not sure what the "fuss" is.  Is TRPF really so heavily disputed?  In a sense, profits are just a kind of arbitrage, and so the reasoning behind TRPF is similar to the reasoning around zero-arbitrage assumptions (profit-maxing agents under perfect competition with free entry/exit and full information will drive supply up and prices down until the profit gap closes). 

Obviously, these assumptions make for boring problems; the real world is far from boring, so profits happen and can be sustained for a really long time in some cases.  I think the diversity of viewpoints you're seeing are more explanations for the varying degrees of failure of TRPF in real markets, than they are explanations of empirical evidence of TRPF.