And once again with that assumption included your statement is either wrong or misleading without specifying what observations exactly together with Hawking theorem prove
that there was a big bang singularity. Have you discovered a new theorem? Please share it with us.
what mistake? Since you are so careful with what is explicitly stated, If you read again my post I don't explicitly say that is restricted to FRW models but that the 1970's version proved that the FRW models must have an initial singularity and then go on to say that it only applies to a certain type of GR solutions, which is true.
BTW, have you already recognized your confusion about what the singularity theorems prove or not prove?