I interpreted the card as leaving the game state in the limit of the game states after n=1,2,3,... plays. As long as that does not require an infinite number of decisions, that is a perfectly well determined thing (or not, depending on whether the limit exist). The limit of playing Monument infinitely can be defined rather well as having infinite $ and infinite VP, as specified by the FAQ.
However, the limit of things with decisions require more careful analysis. Infinitely playing Chapel or Courtyard requires you to do infinitely many decisions, and that is a little harder to define. However, if we consider two sets of decisions that yield the same limit as an equivalent class, there are finitely many decisions, because there are finitely many game states. Thus, the game with ITR can stil be regarded as a finite game, where some quantities that previously could only be integer numbers (like the amount of VP or money), can now also be infinite.
The only problem with those rules is having infinite coin tokens, because you can play those, which means you can end up subtracting infinites. But, as long as we do not play with coin tokens, everything sounds sound.
Of course, adding infinitary limits complicates the game a lot, and I would not use this for anything other than a brain teaser. But I don't think using infinite as an object is as problematic as some of you seem to think.