The problem with analyzing programs in the context of a fixed memory limit is that you would have to revisit all of the proofs whenever the fixed limit was increased. If you can prove something true for unlimited memory, it's also true for any limited amount of memory.
>>we often find it convenient to analyze it as Turing complete anyway
It's convenient, because the proofs will still be true even if memory sizes grow by 500x. Turing machines clearly don't have a real, material existence, similarly to real numbers (assuming that all actual numbers are finite). Real numbers can be considered as modeling integers that are arbitrarily large.
>>we often find it convenient to analyze it as Turing complete anyway
It's convenient, because the proofs will still be true even if memory sizes grow by 500x. Turing machines clearly don't have a real, material existence, similarly to real numbers (assuming that all actual numbers are finite). Real numbers can be considered as modeling integers that are arbitrarily large.