It is kind of normal that 1 bit is used for the sign. So 23 bits + sign is kind of normal. Chances are the noise of the ADC is high enough, that there is very limited value to more resolution. With long time averaging a slightly high resolution can be possible, but this does not need the numbers from ADC chip itself.
The ADA4528-2 OP is not per se better than the AD8629 - they are different: the ADA4528 is very low voltage noise, but higher bias and much higher current noise. The ADA4528 is good for signal sources with less than some 10 K impedance. However the AD8628 is much better for high impedance sources, like the input divider for the higher voltages.
In case one really want's to get sub 100 nV resolution, it would be more like an extra external preamplifier based on an OP like ADA4528. This could also add more gain and have a higher precision protection part (though possibly lower maximum voltage, more bias). Chances are in the lower cost meters the protection is just a resistor, maybe PTC. This could add enough resistance and noise so that it does not make sense to use the ADA4528.
Similar the LTC1151 and AD706 are good for different applications. So it depends on the use. AZ OPs can add quite some spikes, that might upset other circuit parts.
If resistance measurement is more stable than voltage and current measurements, this indicates that the reference noise is a mayor contribution. This might also be the external reference. A lower noise reference may be one of the few actually possible hardware mods, especially if the reference noise is really that bad. Usually the voltage readings near 0 V should be not effected by reference noise, but a reading near to the full scale (e.g. 2 V range, maybe 20 V range) should.
The chip SD ADCs are usually more like good for the lower end 6 digit meters, as the INL can be relatively high. The other problem is the limited range, so that there can be additional gain errors from resistors. So one may get more resolution (low enough noise), but the accuracy is still limited. These SD ADC chips are good for 5.5 digits, and they can even have a good point: the bandwidth for the input noise can be low, as they may not loose time to AZ phases.