Excellent analysis of the problems, brilliantly expressed, right on topic and with some interesting thoughts from philosophy. You know we share much, both quoted Popper, and my work & essay is founded on ontological justification and key points you identify, including; "The history of physics shows that progress in natural science requires a new level of methodology." that "Most of the theories developed do not introduce any new ontology.".
And on maths;"..arithmetization of geometry, as it were, leads to the emasculation of its meaning." 'Mathematics is used in physics only as a method of evaluation, and not as a method of precise calculations. The outcome of this process is neither accuracy, nor understanding,"
also; 'Mathematicians no longer care about understanding, nature or each other.' and; "the problem of the justification (basification) of mathematics is not understood in the conceptual plan and all programs are inadequate."
I certainly agree we need to aim for Cartans; "one axiom, one principle and one material ..point with a vector germ". and work from; "Bodies and forces (as) simply shapes & variations in in the structure of space.
As you've seen, I've shown the veracity in those concepts as they lead to what looks like a classical QM compatible with 'SR'. Declan Trails prrof may be mainly maths but closely matches the ontology in mine & my papers he references (most of my essay was the ontology & experimental analysis). You may not be a QM specialist but if you do perceive any flaws in mine I'd appreciate your comment.
No such flaws in yours, but just one question; Rather more Nature than Maths but I found; "to construct the model of regular process which does not dwell and always lead to something new and new" valid in my ontology. How would a 'sythetic' method be simpler and 'more fundamental?'
Best wishes very well written & top marks.
Best of luck in the contest.