In 5th edition, mounted combat with a controlled mount limits the actions of your mount into the Dodge, Dash, and Disengage actions. However, what about charging at someone and trampling them? Intuitively, it seems like it should be possible. However, I can't find any rules to support this, or any to provide support for a reasonable ruling.
There is a similar question but without an accepted answer of the more general case, in which it is asked if the mount can attack while being ridden. This question is more specific in that I am asking if you can have the mount trample someone while you are controlling it (so it does not have its own initiative).
Best Answer
No, if you control your mount, it can't trample someone. You said it yourself:
None of those includes trampling. You might be thinking that you can trample enemies just by moving through them, but that's not how it works.
Moving through enemies doesn't do damage to them, or knock them prone, or anything else like that. Here's how trampling works in D&D 5e:
This particular example is taken from the Warhorse statblock, but many monsters (such as elephants and mammoths) have similar abilities. Even a mount that has this ability won't be able to use it while being controlled, since it requires them to use their Hooves action, which they can't do.