"We can easily construct a Turing Machine (program) that halts for the first few inputs, loops on the next inputs for a lot more inputs, halts on the next inputs for even more inputs etc. so the chance that it halts fluctuates between 1/3 and 2/3, depending on how many inputs you consider. It diverges rather than converges."

You are supposing an (impossible) uniform distribution in a countable set. Longer inputs have smaller weights.