Recently, it has been shown that the mobility of amorphous InGaZnO (a-IGZO) thin-film transistors (TFTs) depends strongly on channel thickness and metal composition (In/Ga/Zn), resulting in a mobility-threshold voltage (V th) trade-off. To the best of our knowledge, this work provides the first comprehensive modeling study systematically integrating density functional theory (DFT) and machine learning potential (MLP) to capture structural disorder and thickness effects on mobility-V th in amorphous IGZO. We establish the existence of a universal mobility-V th trade-off across diverse IGZO compositions and channel thicknesses. To unveil the origin of the universal trend, we developed a mobility model that covers the full composition and thickness of a-IGZO with composition-resolved parameters using DFT and MLP, taking into account its stochastically and structurally driven variation of the amorphous material's properties. We found that the origin of the mobility-V th trade-off is the strong dependence of both mobility and V th on the carrier concentration. Despite the existence of the mobility-V th trade-off, as a method for designing enhancement-mode devices with high mobility, we propose increasing the Zn content to reduce the structural disorder.