I agree with Bob’s post. In my view, push-hands is training (not fighting) that helps in understanding the opponent on contact. How would you propose Taijiquan training to sense the speed, direction, power, intent, etc. of an opponent’s force without PH (as well as the listening, joining, following, sticking, adhering… as Bob points out)? I think that we need training with an opponent who responds to us with a wide variety of actions so that we can learn proper defense against potential attacks; as well as learning where an opponent is vulnerable to our attacks (including strikes), so that when we decide to attack they are not simply blocked (or partially blocked), deflected, or countered. PH also teaches us how to set up the opponent for an effective attack (including strikes) by influencing their body based on what we sense about that opponent. I would think that the ability to read an opponent, sense their vulnerability to attack, etc. would have enabled the masters of old to land clean, decisive attacks that ended their fights quickly. If we discard the training of sensitivity with PH, then perhaps we are no longer training what I consider important aspects Taijiquan that can be utilized during fighting!
If you are pushed during this training, then you have learned a place where you need to improve since your response was less than ideal. In a fight this could indicate an opportunity for your opponent to strike you. If a push affects you, then it is likely that you would not be able to entirely defend against that same energy when expressed as a strike (i.e. faster, with fajin…). Likewise, if you easily push a training partner during PH, then it is likely that you could have struck them under the same circumstances. If you push them but only with difficulty, then you probably could have struck as well, but they probably could have at least partially defended against it.
I feel that throws can fairly easily be practiced in the context of PH training. Strikes can as well at more advanced levels, but too early introduction of strikes seems to lead to increased tensions if the players are not sufficiently advanced, resulting in a decrease of sensitivity, which to me contradicts one of the major objectives in PH training.
I find that those who do not develop the sensitivity of PH training are less skilled in utilizing the techniques in the two person sanshou set. While, as you point out, the sanshou set also trains certain aspects important in fighting, I feel that freestyle PH, where the emphasis is on sensitivity, trains important skills that complement fighting that may not be advanced with a choreographed sparring set.