Height-offset variables (HOVs) provide a mechanism, known as "pinning at infinity", to lift gradient Gibbs measures (GGMs) - describing interface increments - to proper Gibbs measures that describe absolute heights. Starting from Sheffield's seminal framework, we study HOVs for nearest-neighbor integer-valued gradient models on regular trees, under broad classes of transfer operators requiring only finite second moments and without assuming convexity. We first establish the existence of HOVs as martingale limits, prove the infinite differentiability of their Lebesgue densities, and demonstrate exponential concentration for the associated pinned Gibbs measures. Next we uncover a fundamental trade-off, as the Gibbs measures arising by "pinning at infinity" paradoxically lose several desirable structural properties. We rigorously show that they lose tree-automorphism invariance, the tree-indexed Markov chain property, and extremality within the class of Gibbs measures. Our analysis relies on martingale theory, novel past- and future-tail decompositions, and infinite product representations for moment generating functions, and it applies to free GGMs, as well as to GGMs of height-period two.