Dear Sir,
I thoroughly enjoyed your basic research.
You have addressed one of the fundamental flaws of modern physics. Four years ago, I had written a paper here pleading on the same lines. My this year's essay is also on similar lines. Both our views have much commonality, though explained differently.
Language is the unambiguous transposition of one's/a system's thoughts/command on another person's/system's mind/CPU. Mathematics does that with numbers. Hence mathematics is a language of Nature. But it only depicts quantitative aspect of Nature - scaling up or down the numbers with consequential changes in other aspects. It does not cover all aspects, though. Hence what is true for mathematics may or may not be true for physics or biology. For example, if by paying $3000 we can get a bike, by paying $1000 we can get 1/3 of a bike. This statement is mathematically correct. But physically it does not make any sense. There is no equation for the observer, but it has an important role in physics. Thus, physics beyond mathematics cannot be denied. Extending the limited scope of mathematics (scalar numbers) to (vector) physics or biology, makes us fall in the trap of reductionism. As you say: "mathematics (developed by physicists) that was subsequently abstracted and generalized by the mathematics community" or vice versa.
You have hit the target precisely when you point to the modern practice: "to adapt theories to available mathematical constructs and abstractions however flawed or inappropriate they may be". I can add the process of renormalization and brute-force approach to your list. However, I beg to partially differ from your view on infinities. You can refer to my essay here for details.
Your views on irrational numbers matches my paper, where I had brought out the historical perspective. Similarly, your views on "Algorithms in Addition to Functions" complements my view.
Regards,
basudeba