The answer, at least with ArcPy Data Access cursors, is that you cannot avoid the behavior you are seeing because of a known defect with ArcPy Data Access cursors.  From [BUG-000091314: When using the data access module and an update curs..][1]:

> # Bug BUG-000091314 # 
> ## Synopsis ##
> 
> When using the data access module and an update cursor, row values
> cannot be updated to null.
>
> ## Alternate Solution  ##
> Use the old update cursor or use the Calculate Field geoprocessing tool.

This defect was logged in 2015, which means it was logged right after Pro 1.0 was released, and the defect still exists today in Pro 2.9.x, so I would not be holding your breath for a resolution any time soon.

## **UPDATE**
Esri finally addressed this defect in Pro 3.1 through the introduction of a new `explicit` parameter in the ArcPy DA [InsertCursor - ArcGIS Pro | Documentation][2] and [UpdateCursor - ArcGIS Pro | Documentation][3].

Modifying the OP's code using the new parameter to force nullable fields to accept NULL:
```python
with a.da.UpdateCursor(table, field_list, where_clause, explicit=True) as cursor:
    for row in cursor:
        for i in range(0, len(row)):
            if row[i] == 0:
                row[i] = None
        cursor.updateRow(row)
```

  [1]: https://support.esri.com/en/bugs/nimbus/QlVHLTAwMDA5MTMxNA==
  [2]: https://pro.arcgis.com/en/pro-app/latest/arcpy/data-access/insertcursor-class.htm
  [3]: https://pro.arcgis.com/en/pro-app/latest/arcpy/data-access/updatecursor-class.htm