I agree, this is the crux of the issue.
> It refers to the process of arbitrarily unbounded enumeration (having the potential to count to any arbitrary number).
This is too narrow. The world is full of infinite processes including your life or the earth revolving around the sun, etc. These are the facts that give rise to the concept though enumeration is the archetypical example because its so easy to see. The danger is forgetting that that someone (or a TM) must be doing the enumeration and eventually he (it) will die, planets will be engulfed by the sun, etc. So no real process goes on literally for infinity.
>Aristotle's argument is that the human mind (Turing Machine) is restricted to computing decidable problems and cannot compute undecidable ones.
He went even farther than that -- he argued that what ever the mind (or Turing machine) is processing, becoming aware of or knowing has to be finite too. His basic principle of existence is that whatever exists must have identity, including the mind.
> It does not have to physically exist, but we certainly can theoretically associate a finite characterization to it.
Here is where we disagree or maybe misunderstand each other. You are equivocating on "it". "It" what? By definition, infinity leaves undefined the length, life or extent of the process. Whatever the process is, it is open-ended. Now you can't add back or sneak in some finite measure of the length or extent of the process without destroying its meaning. What you can do is say things about the process even while abstracting away its metaphysical and eventual termination. This is in fact the main value of infinity as a mathematical concept, such as with limits.
> A similar argument would lead to the conclusion that pi does not exist...
Well, what do you mean by "exist"? Clearly the relation it identifies exists but it does not denote a specific number. Pi denotes a specific open-ended process to calculate a number based on what you are trying to do with the math. So if you are buying tile for your circular patio you might use 3.14 to determine the area and how much tile you need but if you are a NASA engineer and want to land a rover on Mars you'll need to carry a few more decimal places, but not an infinite number of them, and the meaning is clear.
That Pi is irrational means that curved paths are incommensurable with linear measures. This did not stop the mathematicians as they just defined a symbol to represent an infinite process (implicitly at first but later fully developed in calculus) that defines the ratio and treat it just like any other number in further equations and theory, and in that sense it is a specifically defined number, and math theory could proceed.
The mistake that mathematicians made, primarily due to Plato, was thinking that Pi is a "completed" number in some super reality where ideal, abstract math exists. It was just a matter of time before Cantor came along and applied the same idea to infinity itself.