Is it generally expected in martial arts systems that all training
stances/positions (horse stances, front stances, drop stances, etc.)
will have direct fighting applications?
Generally expected... No.
In reality. Yes. All stances have fighting applications. All movements in kata or forms have direct fighting applications.
They may not be immediately obvious depending on the type of fighting that you are familiar with.
Horse stance has a couple of applications. It drops your body weight, for example if you have someone in a standing arm bar it puts pressure on the arm and drives them down. It can also be used to break someone's balance.
The problem with the way that stances are taught today is they are seen as static. i.e. a way to stand... It's a profoundly incorrect view of what they are. They are not static, you would be in stance for a fraction of a second. Taking stances out of application, out of context is a good example of poor teaching.
Here's a great example of Machida applying horse stance. It's an application taken directly from the kata Pinan Nidan and one he's gone on to use in other venues.
Application of zenkutsu dachi forward or lunge stance and kiba/shiko dachi horse stances.
Add in a belt and you have obi-otoshi.
Without seeing the movements in the respective forms they're taken from, the first and second examples you give are the same application. Catch a kicking leg, lift it up, push opponent over backwards. The second example it looks like he's also being pushed backwards over the extended leg, there are a number of ways to complete it.
Here is a video example of exactly this application:
Looking at the text of the second question example, the teacher was obviously told what the application of the movement is; "push opponent away", "block a kick" but failed to understand or pass it on.
Is it generally an expectation that there are such applications, and
the stances/positions are trained to achieve those applications?
No, in my experience, the general expectation from martial artists as conventionally trained today (in many schools and systems) is that stances have no purpose and that kata or forms are "dances" which have little to no value.
Personal opinion: I see this as a failure of training and understanding, rather than a failure of the originators of the forms. The quality of teachers of systems which teach martial arts based on solo forms is on average abysmal.