Is that how it goes now? It used to be infinite loops (e.g. playing a Faceless Butcher when the only creature in play is a Faceless Butcher that exiled another Faceless Butchers) resulted in a draw, not a loss.
I suspect most of the MTG infinite loops are something like "A triggers B, B triggers C, and C triggers A". Asserting that the loop will never resolve is not a violation of the halting problem.
The rule cited is worded to only apply to loops that go on indefinitely (machines that don't halt), so the referee can't even know whether or not they're allowed to apply the rule against a without knowing if it halts. On the particular machine mentioned above that would require solving the halting problem.
And yes, I'm sure the people saying "they'd just kick you out if you tried this in a tournament" are right. That's really not the point. The point is the cited rule doesn't really make things better with respect to turing completeness. If magic the gathering without the cited rule is turing complete, then with it it's both just-shy-of-turing-complete* (you'd be able to run any program that does halt) and uncomputable.
* Depending on the precise definition of "turing complete" it might in fact be turing complete. You can evaluate any program. Either it does halt and you can report the output, or it doesn't halt and you can report that you're in an infinite loop because the referee-oracle said so. That's probably enough to satisfy most definitions - even if it's a strange way to implement it.
It's possible that there's a way to do something turing-challenging or busy-beavery in such a situation, but it's usually pretty difficult to make complex unbendable loops, they're usually very direct cases where taking an action more or less forces you to immediately take the same action again, so the loop has...two steps. Anything much bigger or more complex and you're basically assured to have player choice.
There's probably room to do something where a player is clearly losing unless they do something which might continue a situation that is undecidable.
No - because the referee doesn't have to devise a program to do it. They can decide whatever they want. Also the statement of the halting problem isn't about it being impossible to tell whether a program will terminate, it's about it being impossible to devise a program that can tell whether any other program running on a turing machine (which actual physical computers are not, since they have finite state) will terminate.
For any program running on actual physical computer hardware it is obviously possible to tell whether it will terminate. For an actual game of magic, which has considerably less possible state, it's likely even trivial 99.9% of the time. The remaining 0.01% are up to the judge. In the case of MTGA on a computer, there are hard-coded limits on many things (but a lack of enforcement of the first part of the loop rule. Nexus of Fate wasn't a good time.).
If you could, then I'd start using your "real computer" halt checker to instantly mine bitcoin or break all cryptography, so let me know when you figure out out.
