It will be harder to use the opener if there's a three-batter minimum. Managers will presumably toggle their 1-4 hitters aggressively (alternating L/R or moving up their SHs).
It would be fun to see someone break down the numbers on this. How many 1, 2 and 3-batter appearances by relievers were there last season? We know pitching changes are up but how frequent are actual one-out appearances (and how often did they go in tandem with a two-out guy who didn't pitch another inning)? How many batters did Tampa's openers generally face? (At the time of this article they had 50 starts by eight different pitchers for just 93 innings total).
I do agree that it's unlikey the rule will actually lead to fewer relievers and that many fewer pitching changes, because middle relievers will still fails to get outs. Like, say you've got Adam Morgan in to face 3 batters, two of whom are lefties. He gives up a hit to the righty and walks the second lefty. He's out of the game and there's still only one out. The next guy comes in and gets two outs, but he maybe doesn't start the next inning.
If the rule makes a manager stick with his starter too long that probably won't be fun either.