Your title seems to disagree with your point.

Melodrama:

A sensational dramatic piece with exaggerated characters and exciting events intended to appeal to the emotions.

If games were melodramatic they would, by definition, have a plot that was much more than a reason to search a dungeon. Soap operas are melodramatic as are many movies (especially romantic comedies). There isn't anything wrong with melodrama so long as it is a deliberate choice on the part of the authors. Unintentional melodrama is bad. Bad melodrama is also bad (ever laughed when something was supposed to be tragic? That's bad melodrama, it failed to elicit the intended emotional response.)

The thrust of your post seems to be that game plots lack any real plot. To that I say, "Which games?"

Video games, as a medium, are still trying to find their voice. They haven't landed on the best way to form a narrative with an interactive experience. In recent years games have really been trying to recreate the summer blockbuster method of storytelling and for some games that really does work, to an extent. Other games have been experimenting with more subtle narratives, look at Journey or Flower and maybe Datura.

Game designers seem to be struggling with creating games that have a strong narrative while maintaining the interactivity. I don't think better writing will solve this problem. There are so many things that don't translate easily to video games, pacing or reconciling game play with narrative for example. I don't think this is an issue that is easily solved.