The halting problem does not forbid us from discerning if SOME programs will ever halt. It says that there are definitely programs for which we cannot know.
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.
Absolutely, but assuming that magic the gathering without the rule about loops is turing complete, then it's possible to set up all programs in magic the gathering, not just nice ones. In particular there exists a program where it is impossible to tell whether it is in an infinite loop or not without solving 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.
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.