Complaining that reed relays are "rather expensive" is not a valid argument if you want to build a "Precision instrument". Accuracy & quality costs money.
Some of the ADC converters have built in PGA's and MUX, some also have a built in voltage reference.
Even such a chinese IC as the HX711 has two inputs (although not with the same capabilities).
ADS1118 has two input channels PGA upto 128x and a voltage reference.
Those are just two examples. You have to find an IC that fits your needs.
The MUX-es that are built into the ADC's have been matched to the ADC, or at least the errors they cause are a part of the spec of the ADC. If you look at the datasheet of the ADS1118 for example then there are no specs at all for the voltage reference itself. Those are specced as part of the ADC accuracy itself.
When using an external mux, it is indeed "better" to use one with a low on resistance, but you have to consider the whole signal path. For low bandwidht applications it is common to put a capacitor right at the input of the ADC. The ADC usually has a very high input impedance (Giga Ohms) but there is a sample capacitor that gets connected during, well, sampling

and that capacitor has to be charged. If it gets charged though your MUX resistor, there is a voltage drop (charging time may be too short to fully charge it). A buffer capacitor at the ADC input can deliver a current peak during sampling Even without loosing too much charge, and then recharge during the time the ADC is doing it's AD thing.
Another thing you should be aware of is that the sampling capacitor is usually shared between all the channels. If you measure channels with much different voltages, then the sampling capacitor has to be charged and discharged between those voltages each time. If you however only measure one channel, then the charge on the sampling capacitor does not change much at all.
A fun and useful experiment to do yourself is put any microcontroller with and ADC (and MUX, but they all have that with their ADC) on a breadboard, and put a 100nF film capacitor on one of the ADC inputs. Charge the capacitor with some intermediate voltage from a potentiometer, and then remove the pot and watch how the ADC value starts changing.
Then add a second channel and connect that directly to the potentiometer, and sample them both.
You will probably see that the ADC channel that only has a capacitor as input will drift towards the same voltage as the potentiometer, because it gets charged or discharged a bit on each ADC cycle.