I have always thought that without the tee shot relationship, Par 3 holes were good places for "concept shots" like the Redan, or other cross slope concepts, the Biarritz, the precision green (although this one works almost as well on any shorter 4 or par 5 as well. With multiple tees and the ball on the tee, however, the precision green make more sense on a three par), etc.
I tend to favor the precision/small green on par 3 over 210 yards from the back tee to challenge long iron play of better player. I would even admit that an island green is pretty good precision test on a short par 3, even if wildly overused (I haven't done one in 20 years)
I also feel long par 3 (maybe over 240 from the back) holes are good places for partial cross bunkers 20-30 yards short of the green, if we can arrange the tees so most players are using fairway woods that need some roll out. (think Shinny No. 2 or similar)
A multi plateau green works well on par 3's simply because they get more ball marks, so the plateaus give you small targets within a bigger, easier to maintain green. I occasionally do the ultra long or ultra wide green on a par 3, as well, with shallow but wide seemingly another good green for a medium to short par 3, for reasons the plateau green is. However, for some reason, the really big green, like 18 at Indianwood doesn't seem to hold interest on a par 3 as it might on another type of hole.
The "hit it for the fat part or go for the pin" strategy almost seems wasted on a par 3 as it is a basic staple of other holes.
Maybe I am like an old football coach, who over time starts to favor the down and out on third and long, based on experience. But, I like the topic, since after many years of thinking about such things on airplanes, I agree that it seems that in theory, certain types of holes work best on par 3 (or short par 4, long par 4, etc.) Also, in certain wind conditions.