Thus, if I understand correctly, a conformal transformation (which is a coordinate transformation that changes the metric only up to an overall factor) should also be a general coordinate transformation.
Not true. Under a general coordinate transformation, the metric stays the same in an abstract sense, but in order to represent the components of the metric in the new coordinate system, we have to change those components. If we follow all the tensor transformation rules, then the metric gives the same results when we use it to measure things, e.g., $u^iv^jg_{ij}$ gives the same results in the new coordinates as in the old ones.
A conformal transformation is a change in the metric itself. It is not equivalent to a change of coordinates.