Hi,
a comment about the specification:
Output to be expected 0V ... is not a valid requirement, since no OPAMP can fulfill this (unless it has negative supply voltage). Even a Rail-to-Rail-Output-OPAMP just comes close to the rail.
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When using "Offset" to pull up the input, then - if you want some decent accruacy and precision - don´t rely on the power supply. A power supply is neither accurate nor precise. It is expected to be noisy and will drift with time, temperature, load current ...
All this errors you will see at the OPAMP output. maybe even amplified.
--> better use a voltage_reference_IC.
--> Give your requirements as numbers
Klaus