The spec will tell you how the PGA will affect its accuracy. Without a spec we'd just be making up numbers and it's pointless.
Sigma-delta ADCs that I know have increasing offset errors if the source impedance is not low. Works great with low impedance shunts. On the other hand the output impedance of say a Pi-filter or a microphone may be a problem, in which case an amplifier is necessary. Programmable gain may or may not be helpful.
This information is very interesting since it confirms my preliminary hypothesis, that amplification in such amplifiers is made by change of frequency. Otherwise I can't to explain so strange sequence of amplification factors (1, 2, 4, 8, 16...).
But where it has been written for sure???