If you want a constant current load, design a constant current load. As you say, this is a circuit that draws the same current over some range of applied voltage. That's your spec right there.
There are various ways to achieve this. The conceptually simple is to get a measure of the current, then use feedback that adjusts a pass element to keep the current constant. Some current sinks do work this way, especially if a separate supply is available to power the control electronics. For example:
All current entering at SINK goes thru R1. The voltage across R1 is therefore proportional to the current being sunk. Due to how the feedback is arranged, the opamp adjusts the gate of Q1 to maintain the same voltage across R1 as Vref. Vref is therefore the control input that sets how much current to sink.
There are details of stability, bandwidth, and accuracy beyond the scope of a simple answer here. However, such circuits do work and are used in reality. In this case, you'd want a separate power supply for the opamp in most cases.
Choosing R1 is a tradeoff. You want it large for good signal to noise ratio, considering the opamp offset voltage as one source of noise. On the other hand, you want R1 low for lower minimum voltage the current sink can operate at.
Another less accurate but much simpler current sink is:
This exploits the property of a bipolar junction transistor where the collector current is largely independent of the collector voltage over a decent range of collector voltages. As before, all of the sink current flows thru R1. In this case, the base current also flows thru R1, but that's small compared to the collector current due to the gain of the transistor. For example, if the transistor has a gain of 50, then about 98% of the current thru R1 is the sink current and the other 2% is the base current.
The voltage on R1 will be Vref minus the B-E drop of the transistor. At first approximation, that can be considered constant, around 700 mV. This kind of current sink is also used in real life when a few percent accuracy and some load on Vref are OK. This is the case often enough to make this a useful circuit.