Then C-the-language can be Turing complete, even if C-as-actually-implemented is not. Just implement a python interpreter. (Or you can also just implement bignums in C and use those for your computation)
EDIT: See the GMP library, which states "There is no practical limit to the precision except the ones implied by the available memory in the machine GMP runs on"[0]
Yes, this is my entire point
Why should I care what the language specification states in a computability theory discussion? There only needs to exist a method to accomplish our goal-Whether the method conforms to specification or not doesn't seem relevant to me.