Suppose that I have a restricted Turing Machine - it has finite tape and takes bounded input. Consider a program that halts on every input (which is at most $k$ bits). The set of inputs is finite, therefore the function describing running time is bounded and hence is $\mathcal O (1)$.
Therefore, when analyzing any algorithm that has been tuned to support some machine requirement (ex. input is 32 bit integer), if it halts, we may say it's $\mathcal O (1)$.
What am I missing here? What are arguments supporting/opposing this point of view?