Early cars did have VIN stampings. It started with the 1968 model year. I will show you examples if you like.
Stop listening to whomever told you that.
Casting date by oil filter is a good indication, engines could be stamped on intakes, heads or blocks. You will need to check all three.
The transmission stamping is on the left, upper ear of the case. It would not be visible when installed in the car. In the photo the engine would be in the top of the picture. The transmission input shaft is pointing towards the top in photo. The top cover is show (with bolt)