Here are where I have some strong opinions...
In order to be able to fight with Kung-Fu you have to practice it in a way similar to a fight. Full contact being best, sparring being next, and applications or other stuff lest effective. There is no way around it. Kung-Fu is designed to teach fighting, no question about that either.
However, people learn Kung-Fu for different reasons. If fighting isn't one of them that is fine, but you will never truly understand kung-fu or be competent to teach it (** Answer to your question **). If you only practice "applications" you end up with a bunch of BS that involves catching a fist, stepping behind opponent, tieing their arms together then (and so on).... In other words, stuff that looks great in application practice is worthless and will get you hurt in a real fight, but without realistic fight training you will never realize this.
Push hands of any type is a step on the way to real fighting with Tai Chi. It is in my opinion a required step and should not be ignored. However, it should not be your end point in Tai Chi, you need to contine on still.
You should suspect anyone who tells you differently, they are probably trying to pull something over on you.
There are exceptions of course. Many Tai Chi schools teach only for "health" reasons for example. If they make it clear that fighting is no aspect then I don't really have that much of a problem with a non-fighting teacher teaching, but they will probably not be as good as a teacher that does fight. An hour with a great fighting teacher is worth about a year under one who does not fight (At least in my experience)