Interpretations of QM and measurement have very little to do with whether space-time is discrete or continuous. The simple fact is that no common QM formulation

Interpretations of QM and measurement have very little to do with whether space-time is discrete or continuous. The simple fact is that no common QM formulation uses discrete mathematics for space-time, and it's unclear if any that does would even work. Also, your link is not a formulation of QM, it is a different theory which makes different predictions (it is a quantum gravity theory). And, per the sounds of the Wikipedia article at least, it is not actually proven equivalent to QM in the regimes where it needs to be ("There is evidence [1] that, at large scales, CDT approximates the familiar 4-dimensional spacetime", or in other words, it is not fully worked out if this is the case).