The same impedance for all 3 inputs is importance mainly because of the switch resistance (some 70 Ohms for the HC4053) and the related high TC. So this is essentially the reason for the 42 K.
The 3458 uses different switches for the signal and references and thus could use different resistors for the input and the reference. The way it is used there (40 K for the references and 50 K for the input) is still not a solution I like - using also 50 K for the references and 25% higher reference level would have given lower noise.
The other point is the same resistance for the positive and negative reference. This helps that an non ideal zero voltage at the input of the integrator will produce the same offset. An non ideal input can come from the integrator offset (less critical as constant), but also from the settling after the references change. A simple 1 OP integrator would have some kind of square wave of a few mV (depending on the OPs GBW and the integration cap). With the 2 OP integrator like used in the 34401 and most modern designs, reference switching causes a peak (some 10 mV range) that recovers to near zero after some 0.2 to 2 µS (depending on the speed of the OPs). The problem is if these peaks have a slower contribution (that may not be visible on the scope) that extends to possibly the next phase. The peaks are OK if the integrator is at always the same impedance at the input. The constant impedance is only important in the run-up phase - the rundown phase has relatively few transitions in a fixed sequence (but variable timing) is thus less critical.
I don't think the 34401 ADC is more like an upscaled 3457. It is quite different in many points - the 3457 ADC actually works with 10 V full scale and the amplifier adds gain. It only looks a little like they may had a 3 V full scale range in mind. Changing the 100 K at the input to 30 K would make the ADC from the 34401 a perfectly good ADC for a 3 V full scale range, like it was common in the old days.
Using 100 K to a 10 V full scale range and 42 K to me looks like an after-though to make the ADC work with 10 V full scale, at the price of more noise. The resistors can actually be a major noise source for a good ADC. For the integrator with it's current input it may help to look at it as current noise. In this view the 42 K does not contribute to the signal but adds current noise. The 30 K resistors from the reference also add quite some current noise.
Using the auxiliary ADC to measure the voltage at the integrator output is fast, but it also has a limited resolution. The scale of the ADC depends on the integrator resistors and the capacitor. Especially the capacitor tends to be not that stable, so that a frequent check of the scale would be needed. So getting 3 digits from the residual charge is already on the optimistic side - it's more like 4 digits from the run-up and 2.5 digits from the µC internal ADC. The resolution from the run-up part is limited by the speed of the modulation. The very fast modulation of the 34401 has 2 negative consequences:
1) the fixed phases in the run-up patterns reduce the input range. In the 34401 only some +-3.5 V (if they would use 30 K from the input) of the +-10 V reference range are actually used. Slower modulation (like in the 3457, K2000) could have allowed some +-8 V or so.
2) It needs a fast integrator. For the 34401 they choose the OP27 for the precision OP in the integrator, probably for speed reasons. For the noise performance the OP27 is a poor choice here, because of the current noise. Other older precision OPs (e.g. LT1001, OP177) are likely too slow.
A rundown phase needs some time, but not necessarily that much. The 3457 needs some 150 µs, my solution currently uses some 200 µs (120 µs with a single ADC conversion), with a possible speed up to the 50 µs range. So for a 20 ms conversion the time lost is not that relevant. For best precision one would likely anyway alternate between a zero and a signal reading anyway. So some time is anyway lost for the input switching and the continuous integrating version needs extra settling to start with.
The rundown, especially in the classical form does not help much with DA. It avoids possible nonlinearity of the capacitor - however most capacitors tend to be very linear (e.g. better than many resistors). The DA of the capacitors usually has 2 contributions: a fast one from dipole orientation, that happens on the 1-10µs scale and a slow one more from internal surface charges on the 1-10 seconds scale. The fast DA part can be a slight problem with the classical rundown with a comparator to stop the slope. However already a slower slope phase would reduce the fast effect as the last 10 µs before stop don't vary that much. An additional waiting time (some 10 µs) can further reduce this fast effect.
The slow DA mainly hides some charge proportional to the average integrator voltage and gives it back later. This effect does not depend much on the details of the ADC. It can be effected by the way the references a controlled during run-up, as this effects the average voltage. The nice point is that the expected DA related error would follow that average voltage curve, so one has a clear signature to look for. Attached is a curve for the average integrator voltage measured for an input voltage range around the center (horizontal units about 42 µV).
A fast modulation and thus less charge stored in the capacitor is very effective suppressing DA. With a reasonable good capacitor DA should not be an issue for a modulation faster than about 20 kHz. The ADT6581 DMM (8 digits) even gets away with only 5 kHz and not very good feedback during run-up.