Thanks! This is very useful advice!
What I meant by "requiring one of the opamps to sink significant current very close to the negative rail" is that, if you look at the schematic, the differential-to-single-ended op-amp that measures the voltage across the current-sense shunt resistor is using 10kΩ resistors in its feedback path, and the inverting input to that feedback network might be close to the positive voltage rail, say 12V, while the single-ended output is ideally millivolts from ground. So you have 12 volts across 20kΩ, which works out to 600μA, which has to be sunk into that op-amp's output.
600μA doesn't sound like a lot, and it certainly isn't going to strain the drive strength of any op-amp IC, but in this context we're hoping for millivolt precision down near the negative rail. The OPA4197 datasheet https://www.ti.com/lit/ds/symlink/opa4197.pdf figure 14, "Output Voltage Swing from Negative Power Supply vs Output Current (Maximum Supply)", shows what you might call a gently nonlinear output impedance roughly in the 40–80Ω range depending on temperature (2–4V at 50mA), which means 0.6mA of output current works out to tens of millivolts (24–48mV using those nominal impedances). Worse, even under no-load conditions, it's rated to swing only down to as much as 25mV from the negative rail (§6.7, "Electrical Characteristics: VS = ±4 V to ±18 V (VS = 8 V to 36 V) (continued)", p. 8, "Vₒ: Voltage output swing from rail, Negative rail").
In retrospect, it seems obvious that the op-amp's output isn't going to be able to reach beyond the input rails (unless it integrates a charge pump like the LM7705 internally) and is going to have trouble getting too close to them when it's sinking any current (for the negative rail, or sourcing for the positive). Because where is that current being sunk to? You need some voltage drop to get the electrons and holes to move in the desired direction through the silicon. A small negative supply might be the right solution. Or a differential output, which would be easy.