Why do LDO regulators have so big a voltage drop?
There are regulators with a drop out voltage close to 0&nbsp;mV. Check figure 5 on page 6 in TPS73101, Cap-Free, NMOS, 150mA Low Dropout Regulator with Reverse Current Protection.

Another example is LTC1844 - 150 mA, Micropower, Low Noise, VLDO Linear Regulator.

The problem with regulators at such low drop out voltages is that in those regions they have crappy parameters (line/load regulation and PSRR).

As to the part if it is possible to build such regulator with an op-amp and a discrete MOS device - yes, it is possible. You will have to use PMOS and take care of stability (it is not easy to make a feedback loop stable in such a configuration).