They had Ned Garver, who won 20 games and was the starting pitcher for that year’s All-Star game.
Unfortunately, the entire Browns team only won 52 games, and Garver was the only pitcher on the staff to have a winning record.
The rules of baseball aren’t static. If teams can find tiny players who aren’t simply making a mockery of baseball the rules can be changed. They can make the strike zone a minimum height and then it’s a 3’7" player will most likely strike out or dribble one into the infield for an easy out.
Even when walking such a player is not an offensive asset. They can’t take a significant lead off first, giving the first baseman more fielding range. Not only can’t they steal but their slow speed will prevent the next batter from getting multiple base hits and make double plays more likely. Hardly what you need from a DH or pinch hitter, which would be the only way to keep this defensive weakness off the field. So at best he’s a pinch hitter in rare situations, taking the place of someone more likely to get some actual bases and contribute to scoring.
Here’s something that’s changed since the 1950s. There used to be proportional dwarfism. It was pituitary dwarfism. People didn’t make enough Human Growth Hormone, and it prevented them from achieving their genetically programmed adult height. There is now an artificial HGH, and anyone whose blood tests reveal them to be lacking HGH are given shots, and achieve their full height. There are no more proportional dwarves (aka, “midgets”). The only little people who are going to be 3’7 are going to have something like achondroplasia, or primordial dwarfism, and that is going to present other problems with playing baseball. Today’s little people rarely do not have problems like scoliosis, or tight foramen magnum, that can cause problems during exertion.
So while you do have little people games, I doubt very much you would have a little person cleared to play in the majors. It’s possible, but doubtful. You do have other forms of dwarfism that lead to taller people, such as hypochondroplasia, where people are often upwards of 4’, and have fewer ill effects.
Then, there are occasionally people who are just really short, who never got treated with HGH, because they weren’t unusually short as children. They might be 4’11, or 5’. They might be excellent athletes-- I know one who was a very successful bodybuilder, boxer, and former high school wrestler. He was 4’11, and that was his genetic height. An inch too tall to be considered a dwarf, and not unusually short as a child, so no one suspected he would be short as an adult. He could run like the wind, and probably could have done great in pro baseball, if the idea had been to have someone with a small strike zone.
Anyway, proportional people under 4 ft. have gone the way of the Dodo.
You’re allowed some substitutions in baseball. Start the inning with enough real fielders to play the game, and maybe accept that you let a few more runs through than you’d like, then swap in all the rest of the walkers in the bottom and rack up 30 or 40 runs, then swap back to other real players in the next inning. Again, assuming you could always walk.
Your ideas are intriguing and I want to subscribe to your newsletter. 
Some of these ideas for an entire batting order full of midgets would actually be feasible in September, when the major league roster expands. Imagine a team that’s in a tight pennant race looking for every possible advantage.
You’re only allowed 25 players on your active roster during most of the season. If you burn the 9 defensive players you used in the top of the first by replacing them with walkers in the bottom of the first, you would only have 6 real players to swap back in in the second.
If you assume a pitcher is occasionally going to be able to strike out a dwarf, the opposing team might be able to rack up a lot of runs the next time they came up.
Not sure how a one-off screwball publicity stunt that didn’t have even a thousandth of the societal impact of the Battle of the Sexes keeps generating so much intense debate, but then I didn’t think the Tim Tebow train would chug along that incredibly long, so I guess you just never can tell.
My understanding was that the powers that be briefly considered instituting a height minimum, then ultimately decided that it wasn’t necessary and would be seen (rightfully) as a gross overreaction. The problem was that Gaedel wasn’t a legitimate player; no more, no less. There’s nothing about a too-small strike zone that can’t be taken care of with the sport’s existing practices. Pitchers can learn to adjust to short batters, and umpires can be as tight or generous as they want. Bottom line, if all a player has going for him is that he’s very hard to strike out, he’s not going to last long, especially in the modern offense-oriented game where one run usually isn’t enough.
For the record, the only sport that does have a height minimum is sumo. The Japan Sumo Association requires all new recruits to be at least 1.73 meters tall. In the past a few applicants actually used temporary scalp implants to make the height; this has since been banned.
1.73m is about 5’8” which seems like a tall cutoff but I don’t know much about sumo. Do they have a sumo league for shorties?
Yeah, you don’t need much intrusion from reality to make the idea utterly impractical. I was just speaking of the world of ideal abstractions.
Besides Garver, there were a couple guys who had good careers in Dale Long and Roy Sievers. Tommy Byrne was an effective pitcher (and hitter) as a Yankee but walked a lot. Maybe that’s why he was effective with the Yankees. Casey Stengel put emphasis on turning double plays and avoiding them. Veeck wrote in his book that the player Gaedel hit for, Frank Saucier, felt humiliated by the events and never developed. Tough to tell by the data, if complete on baseball reference, shows him as a good hitter in the minors and nothing after the Browns 14 at bat career. Sherm Lollar, the catcher, played until he was 38 and was on several All Star teams.