If all the various oddities folks seek examples of on this site, par-3 opening holes seem to be the rarest.
I would suspect that this is, at least in part, due to pace of play issues with tee time intervals.
It's a subject of great interest to me right now because UNC Finley (Cobb/Fazio) is going to undergo a major project beginning in October by Love that will, among other things, have the current par 3 12th hole become #1.
As an experiment, I kept a clock on my group the last time I played there; three of us, all single digit. My two partners both hit the green; one made birdie, the other had a tap in par. I missed in the fringe, and had a relatively easy two putt par. So a total of 8 shots, 5 of them with a putter. Nobody in the deep bunker on the left, nobody in the hazards on the right or behind. It took us exactly 8 minutes. So that's good, right? We were two minutes below the 10 minute intervals between tee times...
Now imagine a foursome in which nobody hits the green, and their tee shots DO end up in bunkers and hazards, and each player has to hit a second shot with a club other than a putter. I think it's possible that it could take that group close to twice as long as it took us to play the hole, and that's probably a much more typical outcome, especially on the first hole of the day, than what happened to us on our 12th hole of the day. And we won't even get into groups that are going to hit two off the first tee.
Get a handful of foursomes that take in excess of 10 minutes to play a par three opening hole, and your tee times are screwed up for the entire day. So I suspect that a par 3 opener would only function well at a course that didn't get a lot of play to begin with, and/or doesn't have tee times at all.