Should relief pitchers be forced to face more than one batter?

Recently it was observed that MLB could speed up games by cutting down on the number of pitching changes, each of which lasts around four minutes, and which there have been more and more of in recent years. One way to cut down on the number of pitching changes: Create a rule that any pitcher who enters the game must face at least two batters.

The one-batter relief appearance used to be a rarity. In the 1920s and 30s there were something like 60 total appearances a season (among all 16 teams). In the 40s and 50s the total crept up to ~120 a season. By the 1980s, with bullpen usage having changed some and a number of new teams having been added, the number was up to ~500 a year. In 1992 the previous record was blown away and there were 777 one-batter appearances. In the last ten seasons there have been at 1,038 such appearances every year, with the three highest totals ever coming during the last four years. (The current record of 1,326 was set in 2012.)

The majority of those one-batter appearances involve a left-handed pitcher being brought in to face a left-handed batter, but there are plenty involving righties as well.

The single-team record for one-batter pitching appearances was set by the 2004 Giants, who had 95 of them. The American League record was set by our very own 2014 Indians, who had 80 of them (Marc Rzepczynski had 23 of those, while the single-season record for a pitcher is 37, set by Randy Choate in 2012.)

Would such a rule be feasible? In short, yes. Almost any rule is feasible, if baseball wants it badly enough. There would have to be some sort of language in the book for when a pitcher is injured (or "injured," because managers would surely make such claims if there were no consequences), but that sort of thing could be worked out.

So, such a change is possible, but is it really a good idea?

If the only reason for the change is to save time, I'm not sure how worthwhile it is. Yes, the average pitching change lasts almost four minutes, and yes, the number of pitching changes keeps going up, but the number of one-batter appearances still works out to roughly one every two games, which means you're going to shorten the average game by not even two minutes.

If MLB makes that change, I think it would have to be because there's a sense that it would be better for baseball in ways that go beyond saving 100 seconds or so a game. Presumably such a change would boost offense, something I'm certain MLB would like to see happen (even though they've wrung their hands about the last offensive explosion for roughly a decade now).

It's hard to know what all strategic implications such a rule would have. Relievers with extreme platoon splits would see their value hurt substantially, to the point where a few might lose their roster spot. Managers might choose to operate with a smaller bullpen, but maybe pitchers would be less capable of pitching on consecutive days, and so a larger 'pen would still be needed.

I tend to be against dramatic changes, but I also don't think "Because that's how it's always been" is a compelling enough reason to end a discussion. If MLB is looking for something else to try in the Arizona Fall League, this might be a good one, though a short season with such young players might not be the best laboratory for such a thing.

What do you think? Is the idea worth exploring? Why or why not?