The ellipsis has a few uses:
For all but the last, it's usually easy to tell from context that it's being used for that purpose. So if ellipses start showing up in places that don't fit any of those situations, to me it sorta defaults to the last one, which is the "sad" tone your friend mentioned.
A period, on the other hand, is pretty neutral—it only seems curt if you write exclusively in ultra-short sentences. Generally, avoid doing that. It's rarely necessary. It's often awkward. It's sometimes the right choice though. If not, don't use ellipses. Use conjunctions and dashes. String sentences together. If not, don't use ellipses—use conjunctions and dashes, and string sentences together.