I think you should start with a beat sheet; like Blake Snyder's 15 beat sheet, or Noam Kroll's 40 beat sheet, which expands on Snyder's.
You don't have to hit every beat, of course, especially for a short film, but films do roughly follow this outline and timing.
When you are done with the Kroll beat sheet, each beat is 2-3 pages in screenplay format to get to 100-120 pages of screenplay. Adjust and eliminate to get down to your short film length. You might eliminate most of the second Act, for example, or condense it to a few pages.
The "fun and games" section may be very short.
The problem with discovery writing (which is what I do, finding the story as I go) is that it is not well suited to the tight timing required in a screenplay. If you are "discovering" your story, it is best to do it in outline form (which is what beat sheets are). When you think you've got your scenes down, it is easier to write those sections, and they will tie together. You can even include the specific dialogue or action sequences you want to be sure is in each beat.
To me, the problem with trying to write the full blown screenplay from scratch is I exceed the page limit about a third or halfway through; and see no good way to cut it down. Controlling the length is paramount, so each scene is precious. For me that means thinking long and hard about how to make each scene and every line of dialogue do a lot of work, double or triple duty; something that is not so important in a novel.