Dear Constantinos Ragazas,
Our opinions about 'mathematics' and 'modern physics' converge: 'Writing bad physics using good mathematics leads to counter-intuitive physical explanations', and 'modern physics does not provide physical explanations that make sense'. As you have stated somewhere above, physical realism is necessary, and is possible; and I would like to add, physical realism is the truth.
The existing convention is that we arrive at certain 'mathematical relations', verify these with experimental results, and based on these arrive at conclusions regarding 'the nature of the physical world'. This may appear to be the right path. But mathematics can trick us. A unique mathematical relation can have different physical interpretations, from which we have to select the right one. But from the time of Newton, physicists habitually selected the interpretation that looked 'mathematically simple and beautiful'. They did not care whether there can be other interpretations. The net result is that we have arrived at wrong conclusions that does not make any sense.
What is required is a 'physicalist' approach: Out of the possible interpretations based on a certain equation, select 'one' that has a clear physical meaning. And it is possible to do so. The equations of Newtonian mechanics, quantum mechanics and relativity mechanics can be interpreted in alternate ways to obtain physical explanations having sense. Please go through my essay: A physicalist interpretation of the relation between Physics and Mathematics.
Regarding energy I would like to ask a question: What is energy? My answer is that fundamental particles of matter move at speed 'c', or motion is a fundamental property of matter. Energy is a measure of this motion, and is always discrete. If you feel interested, please visit my site: finitenesstheory.com.