The output of the Remesh modifier seems to lack all mesh data layers, including UV layers.
The Data Transfer modifier requires mesh data layers to already exist in order to work. It thus does not have any effect when placed after a Remesh in the modifier stack.
The "Generate Data Layers" button in the Data Transfer modifier does not solve this, as it operates on the base mesh, and not on the output of the Remesh modifier.
Is there any way to non-destructively re-create a UV layer after a Remesh modifier in the modifier stack? The created UV layer does not have to preserve any coordinates, mappings, or actual data, as that is the function of the Data Transfer modifier. It only needs to exist, even if empty, so that the Data Transfer modifier has somewhere to put its output.