For a NMOS pass transistor, we would ideally like it to be in linear mode, but most of the time we have no choice as for it to be in the linear mode, the gate drive must be at least one threshold above the line voltage.
Therefore, it is in saturation most of the time, where you would need at least one VGS of dropout. The smaller the load current would only result in a smaller gate overdrive.
Lastly, for LDO, unless we have another voltage rail above the line voltage, we usually use a PMOS pass transistor for LDO, as we can easily operate it in linear mode.