MOSFETs do have very low input bias current, but they also have higher noise than bipolar or JFET transistors. In most high-side current sense applications, a very small resistor is used to minimize the voltage drop in series with the load. This gives a small voltage signal that we need to pick off with a low-noise input stage. There is usually a fairly high current we are trying to sense, so a few nanoamps or even microamps of input bias current with a bipolar device will be a negligible error. Take a look at the AD8418A current-sense amplifier. For very common mode voltages, you could try the AD8479.
