Why not use a diode instead of a resistor for current sense circuits?
You can choose the resistor so that it consumes the same power, or less if you want to. At 2&nbsp;A the diode will consume 1&nbsp;W, so "equal" resistor would be 0.25&nbsp;&Omega;. At smaller current, say 1&nbsp;A, the diode will consume roughly 0.5&nbsp;W because the voltage doesn't drop that much. There the resistor only consumes 0.25&nbsp;W, which is significantly less.

The diode is not accurate. Its \$V_f\$ varies especially with temperature. Junction to ambient thermal resistance is 45&deg;C so you can expect that much heating with the maximum current. And if you have some heating from outside (direct sunlight for example) you can easily reach 100&deg;C temperature for the junction which drops \$V_f\$ to half from the original. A decent resistor drifts something like 50&nbsp;ppm/&deg;C, so at temperature rise from 25&deg;C to 100&deg;C the resistance changes only 0.375%

With a resistor you do need an amplifier, because at low current the voltage is also low. But there are circuits made for just that, so implementation is not difficult. Search for e.g. current sense amplifier. Some of those even have a built in comparator so the number of components might even be the same as with the diode + comparator solution. And while you are adding an amplifier, you can drop the resistor size, and get away with a lot less power loss than with the diode.

So the answer to your question is, diode based solutions are not accurate, and consumer more power compared to shunt resistor solutions.