non behaving opamp will give nasty effect such as phase reversal but i think tl071 is not one of them.
No, the TL07x and its brethren WILL show phase reversal when the lower Vcm range is violated. I have a funny* story about TL074s being used in a device that was supposed to monitor a split supply and shut down some associated signals in case of power supply failure. Guess what happened when the negative supply failed or simply came up slower than the positive side of the supply!
* Funny in hindsight, anyway. At the time it resulted in an awkward series of conversations with the manufacturer. . .
In order for the slave amps to measure the voltage drop across the resistor one must necessarily put the ground where I put it.
Whether or not the output signal of an op amp is meaningfully referenced to ground depends on if its input and feedback networks are referenced to ground (and how). The negative rail of the op amp is irrelevant, as long the op amp's supply and common mode voltage ranges are observed.
In your circuit, the slave circuits just sink more or less current through their sense resistors until the voltage at the top of the sense resistor matches whatever voltage the master op amp is providing. They don't care if either or both is offset with respect to ground, because the master's feedback loop is only closed through the current they cause to flow through the master sense resistor, and therefore its output is not meaningfully referenced to ground. The master's output will swing as high or as low as it needs to (or can) to get the master sense resistor voltage to match the master set point. If the master op amp were configured as a simple fixed gain buffer (with DC feedback directly from its output to its input), then the signal at its output WOULD be referenced to ground and the offset would matter. But it's not, so it doesn't, and even then the negative rail doesn't need to be anywhere in particular for it to work.
There is really no such thing as an op amp that cannot be used from a single supply. Dual rail op amp requirements are not really a thing. Although they may not have the performance requirements required to fufill the design, but that's another issue.
Sure, technically, an op amp doesn't care where its supplies sit with respect to ground. Some, however, very much do care about where their inputs and outputs are with respect to their supply rails! That's why op amps are spec'ed with a common mode voltage rating. Saying that an op amp requires a split supply is short hand for saying that it will behave either poorly or even maliciously if you operate it too close to its negative rail. The TL07x happens to be one of the latter. If you won't take our word for it, wire one up as a unity gain buffer, sweep its input, and watch what happens on the output.