There really are, to some degree 'levels of Platonism". "Will we ever find a machine that halts on this program" does not necessarily require infinite resources to answer either positively or negatively. And different 'levels' in the continuum between Ultra-Finitism and true Platonism will give different answers.
For many programs, you can answer the question positively by inspecting the program and constructing one that does. That does not require infinite resources. But it may require a level of resource consumption that is excessive and incomprehensible.
The Ultra-Finitist would put a definite limit on how fast your constructed machine would have to run in order to be considered 'real'.
A mere Finitist would find it sufficient that you can put a boundary, any boundary on how long the machine could run.
A Constructivist would find it sufficient to prove that if the machine did not halt, then some other problem of know difficulty could be solved faster than we know it can.
An Intuitionist would find it sufficient even if the machine could not be directly produced, but if it were the limit point of some progression of more and more complex machines that halt on related problems of increasing complexity, as long as the complexity of the problem increases faster than the complexity of the machine and eventually overtakes it.
A Classical Formalist would find it sufficient if there is a contradiction between the halting not happening and some other definable infinite structure not being complete. For instance a diagonal argument such as the one that proves the reals are not countable. So there would not be a construction, or a general rule for construction, but an infinite list of things that can be defined, and that would not exist were there no such machine.
A Platonist would not even require the infinite structure to be definable in the strict sense, only that its existence should be consistent with logic.
The range of allowed negative answers is similar. A contradiction is a constructed object that contradicts the existence of another constructed object.
Each step on this ladder believes in contradictions that are as materially constructed as they allow their real objects to be. Contradictions that are less clear rule out proving the result, but they do not prove it false. They leave successively smaller gaps of the irrelevant or unknowable between the true and the false. Nowadays, since we have thought up things like Russel's set and the Barry's number, even a sane Platonist must allow for this gap, since clear language still contains paradoxes.
There are rungs between and beside these points as well, but it is harder to nail down precisely what those mean. In particular, modern Set Theory like the theories of Large Cardinals, dwell in the middle ground between Classical Formalism and Platonism, allowing infinite constructions via transfinite induction across all ordinals, but not arbitrary combinations of other ungrounded collections.