Maybe its nostalgia, but I really dig these old game animations as they set the tone and story for what you are about to experience in gameplay with a condensed format. I'm mainly thinking of the old Sims games like Sims 2 especially, I appreciate how the intro is split up into showing different areas briefly and stringing them together. Theme Hospital doesn't reveal much to do with the story of the game but is rather a short comical skit showing how you can be the hero of the hospital but also get away with dark intentions as shown at the end.
Video Links (working as of 2023):
Theme Hospital
The Sims Bustin' Out
The Sims 1
The Sims 2 (Console)
The Urbz: Sims in the City