Depending on the actual HW used in the ADC, in non AZ mode there can be quite some drift and also low frequency noise (from the OPs, but also the resistors) that is suppressed with the zero reading. So it is not only about hours but already seconds can make a difference. So if time permits the zero reading can be really helpful. Doing the zero reading just after the actual signal could be a really good solution, if one can live with the extra delay for the result. However I am afraid this could be too close to the basics to be supported in the current software - so it may need a updated software. Ideally there would be different versions of the auto zero mode to choose: zero before signal, zero after signal and the current version that is likely doing some interpolation / extrapolations on the zero readings over some time frame.
The zero reading seem to be a tricky topic in the Keithley meters. It looks a little like some of the extra noise seen at the 10 seconds time scale is due to a not so good solution in doing the zero readings or taking the readings into account. This is kind of effecting the very basics of the meter, but it could be worth to get a better solution there. It may not be top priority with the DAQ6510 but a similar weakness shows up in other meters too - so kind of a legacy weakness, possibly dating back to the K2001 or even 19x times.