I don't think it's possible for ArcGIS to produce a table without an OID column. Therefore, a solution/work-around would be to remove the OID column after it is created.
You can do this within the model itself by using a 'Calculate Value' item to run some Python code to remove the OID field after the export. (You can just ignore the output of the 'Calculate Value'. Or you could use it as a precondition if the model needs to do anything else afterwards.)
The Python could use a CSV or data management module such as 'Pandas' and would be a fairly simple script, something like the following (untested)...
removeOIDFromCSV("%Output Table%", "%NEW Output Table%")
def removeOIDFromCSV(inCSV, outCSV):
dataframe = pandas.read_csv(inCSV)
Note that the percent symbols in the Expression are used to delimit the names of other objects in the model. When the model is executed, these items will be replaced with the values of those objects. In the example above, you would use the names of your original output CSV object in the model, and the name of the intended (no-OID) output CSV object in the model (or if there is no such object elsewhere in the model, just use the actual full path to the intended CSV location).
Note also that in this case, these parameters must be in quotes, because they will be string literals.