Well yes. We can certainly make a function that acts something like that oracle in some special cases. But my point was to give an example of something that cannot be constructively created. The oracle that I described cannot exist within the universe of constructable things.

You literally cannot doubt the existence of this oracle, without doubting what existence means in classical mathematics.

Here is why a classical mathematician would say that this oracle exists.

Let f(program, input, n) be 1 or 0 depending on whether the program program, given input input, is still running at step n. This is a perfectly well-behaved mathematical function. In fact it is a computable one - we can compute it by merely running a simulation of a computer for a fixed number of steps.

Let oracle(program, input) be the limit, as n goes to infinity, of f(program, input, n). Classically this limit always exists, and always gives us 0 or 1. The fact that we happen to be unable to compute it, doesn't change the fact that this is a perfectly well-defined function according to classical mathematics.

If you give up the existence of this oracle, you might as well give up the existence of any real numbers that do not have a finite description. Which is to say, almost all of them. Why? Because the set of finite descriptions is countable, and therefore the set of real numbers that admit a finite description is also only countable. But there are an uncountable number of real numbers, so almost all real numbers do not admit a finite description.

The real question isn't whether this oracle exists. It is what you want the word "exists" to mean.

Everyone agrees that you can't write an algorithm for a Turing machine (or computational equivalent) that decides the Halting problem for Turing machines in every case. Since this is explicitly worded as "you can't write an algorithm for..." it's in fact talking about a kind of constructive existence and saying that it doesn't apply. The oracle concept is normally phrased about "If we magically had a way to decide this undecidable problem in every case" but it's real utility from a constructive POV is talking about special cases that you haven't bothered to narrow down just yet.

The underlying problem is that constructivism and non-constructive reasoning are using the word "exists" (and, relatedly, the logical disjunction) to mean very different things. The constructive meaning for "exists" is certainly more intuitive, so it makes sense that constructivists would want it by 'default'; but the non-constructive operator (which a constructivist would preferably understand as "is merely allowed to exist"), while somewhat more subtle, has a usefulness of its own.

So having a different philosophy from you makes me presumptive?

What makes you presume that you have any business telling someone with different beliefs from you, what is OK to believe? You may believe in the existence of whatever you like. Whether that be numbers that cannot be specified, or invisible pink unicorns.

I'll be over in the corner saying that your belief does not compel me to agree with you on the question of what exists. Not when your belief follows from formalism, which explicitly abandons any pretense of meaningfulness to its abstract symbol manipulation.