The Greater Caucasus (GC) Mountains within the central Arabia-Eurasia collision zone, are an archetypal example of a young collisional orogen. However, the mechanisms driving rock uplift in the range are controversial, with recent provocative suggestions that uplift of the western GC is strongly influenced by an isostatic response to slab detachment, whereas the eastern half has grown through shortening and crustal thickening. Testing this hypothesis is challenging because records of erosion and exhumation rates mostly come from the western GC, where slab detachment may have occurred. To address this, we report 623 new, paired zircon U-Pb and (U-Th)/He ages from 7 different modern river sediments, spanning a ~400 km long gap in bedrock thermochronometer data. We synthesize these with prior bedrock thermochronometer data and recent catchment averaged ¹⁰Be cosmogenic erosion rates, structural observations, and plate reconstructions. We find no evidence of major differences in rates or timing of onset of cooling or total amounts of exhumation along the length of the range, incompatible with previous suggestions of heterogeneous drivers for exhumation along-strike. Comparison of exhumation and extrapolated millennial erosion rates are permissive of an acceleration of rates in the western GC, but are problematic because these regions are also influenced by Quaternary glaciations that may bias the erosion rate estimates. Intergration of these new datasets with simple models of orogenic growth suggest that the gross topography of the GC is explainable with traditional models of accretion, thickening, and uplift and does not require any additional dynamic explanations.