Electronic – Opamp topology without a stable voltage reference

They are regulators as you thought.

To properly control the MOSFET on each regulator, the gate needs to rise several volts above the source, probably over 6 volts greater than the source. If the source voltage is 3.3 volts the gate might need to rise as high as 9 or 10 volts when the MOSFET is driving a heavy load current. This means the op-amp MUST be powered from a supply greater than 10 volts.

The non-inverting input is the demand voltage for the regulator - for the 3V3 supply the non-inverting input will be 3.3 volts and ditto the 2V5 supply.

The MOSFET is rated at 20 amps and this means the supply current to the circuit might be in the realm of 5 to 8 amps - try and find a linear voltage regulator that would fit the bill an be a low drop-out type - realistically with a drain voltage of fractionally above 3.3 volts the output would still be in regulation. This is of course due to the op-amp being powered from 12 volts.

As to whether it is a bad design, I'd say not BUT I don't know the full spec of the circuit.

EDIT: I didn't see OP's edit where he says he already tried this. Anyway I'll leave this answer up for future readers.

You are correct that most voltage references cannot both source and sink current. One option is to either use a shunt reference or to pre-load a series reference:

^{simulate this circuit – Schematic created using CircuitLab}

In this circuit, normally the reference IC provides 13 mA to R1. If the actual load (connected at OUT) needs to source current, that current will flow through R1 and U1 will just reduce its output current. I've shown both a filter capacitor (C1) to maintain stability of the reference/regulator, and bypass capacitance (C2) to take up most of the switching current and maintain steady voltage at the load under transients.

However, pre-loading is fairly wasteful of power (about 26 mW burned in the pre-load resistor, plus the extra current through the series reference). Andd 10 mA is a lot to ask from a precision reference circuit.

But if ~50 mW doesn't bother you, and you don't need sub-1% accuracy, it should work okay and doens't cost much.