What is the need for amplifier stage in a data acquisition hardware?
1) Even if the IA stage does not amplify, it does provide other functions:


buffer the input signal so that any disturbances present at the ADC's input cannot reach the multiplexer.
provide a well defined input impedance for the multiplexer and/or the circuits connected to the multiplexer.
provide a buffered signal for the ADC. The ADC might require a (small) input current, it might have an infinite input impedance. With the IA in place as long as the IA can drive the ADC's input impedance, there is no issue.


2) The IA consists of opamps with feedback. Such circuits suppress the non-linearity of the circuits (in the opamps) by having excess loopgain.

Let's say that the opamp has a voltage gain of 10000 (10k) at a certain frequency, but you configure the feedback so that the opamp with feedback has a gain of 10 then the excess loopgain is 10000/10 = 1000 which means  a factor 1000 supression of non-linearities. So 10 % distortion would be reduced to 0.01 % distortion.

Also that differential pair will not have more than a few mV across its inputs. So it does not need to work with 10 V or 2.5 V, it cannot handle that anyway. Opamps work linearly only when they have a large gain which causes the input voltage (of the differential pair) to be quite small. For example: if an opamp needs to output 2 V and has a gain of 10000 then at the input the voltage is only 2 / 10000 = 0.2 mV.