So 0V at the terminals may be 140mV at the ADC input and read as 253 but will be displayed as 0uV.
And 10V at the terminals may be 1.140mV at the ADC input and read as 10123123 but will be displayed as 10.000 00V.
Ay, there's the rub.
I'll try to make myself clearer, it seems I was not able to convey my point.
I'll use a 10:1 input divider, and 15 mV op-amp minimum output - this last value is much less than the worst case, but let's go with this.
0 V will be displayed as 0 V - the initial, 15 mV offset has been "calibrated" away.

5 V will be displayed as 5 V - the reading has been matched to a known good reference

All is fine and dandy, isn't it? Well...
Let's see what happens for 149.999 mV at the input, i.e. 14.9999 mV at the op-amp + pin, with 15 mV as initial offset).
The op-amp out will still be 15 mV: that's the minimum it can output - remember we are not (yet) talking about the offset of the op-amp.
So the input range 0 - 150 mV will be displayed as 0 (remember, we have a 10:1 divider!), nice large dead range there

As our 0 point is 150 mV, and our 5 V point is 5 V the line equation has m = (5-0)/(5 - 0.150) ≈ 1.031, and b ≈ - 0.155.
Let's see what happens for, say, a 2 V input.
Now the op-amp is happily working in its foreseen range, so its output is a nice and round 200 mV (let's disregard the op-amp offset...).
Vdisplay ≈ 2 × 1.031 - 0.155 = 1.907

Conversely for a 15 V input:
Vdisplay ≈ 15 × 1.031 - 0.155 = 15.31

I stand corrected, it's not even 3 digits!
The so called "calibration" has absolutely worsened the characteristics. Good thinking.
It would have been better to just calibrate the 5 V point, at least the range 150 mV - max would have been (more) correct.
The solution is not difficult, just provide a small negative supply to the op-amp. Then start to wonder about its offset...
Honestly, given this basic system flaw having guard traces and removing the soldermask in the improved version looks really like lipstick on pig, and cargo cult design.
EtA: The FW does not take into account that the LTC2400 can measure also a 0.125 × Vref interval for
negative input voltage. The sign is simply thrown away when reading the bit stream. So, even if we had an ideal op-amp, the noise will be rectified and averaged to a positive offset.
Shoddy FW for shoddy HW.