But I am wondering if this would work...purchase a Mac Mini to store purchased movies and watch them on my HDTV.
Yes. I replaced my ATV1 with a Mac mini and it resolved all of my ATV frustrations.
Would I even need an Apple TV in that instance?
No.
It seems that a Mac Mini and an HDTV would be all I need to rent or buy movies (thru iTunes) and watch them on my HDTV.
Yes, you can rent or buy from iTunes on the Mini as well as store your entire library on it or any external drives connected to it. You can also watch web video, including Hulu and Netflix, directly in Safari.
The only annoyance comes from the irrational selection of video types in the iTunes store. Some titles are only available in HD on an iPad or an ATV, despite the Mac mini's superior hardware and performance relative to those devices.
Is Apple TV just an option if you don't want to purchase a Mac Mini?
I'm not sure how relevant it is to compare them directly. The ATV is a dedicated media player designed primarily for use with iTunes files, although it also has an embedded Netflix player. The Mac mini is an actual computer with an HDMI port that can do 1080p video output with surround sound. The price difference between the two is substantial, so the ATV makes for an economical, albeit limited player. The Mac mini has none of the limitations or performance issues of an ATV, but the extra power and versatility come at a price.