Bleh! I agree with those who find the beauty of baseball in that (the DH anomaly aside) everyone who plays, PLAYS. In the field as well as the batter’s box.
From a purely practical point of view, we might see the onset of 5+ hour games. Very rarely is a big inning begun by a miscue in the field, as opposed to an ineffective pitching performance, and pitching is already specialized with the DH rule in the AL. If one were able to field a 1-9 lineup of DHs, this would not improve the fielding or pitching nearly as much as it would allow one-dimensional sluggers to hone their art.
To follow suit with football, baseball would have to have a clock or a mercy rule. Both of which are completely against what baseball is fundamentally about: no matter how big the lead is, until the 27th out is made, you have the throw the ball, you have to field the play, and give the other team a chance to swing the bat.
As for the question raised by the OP, I was under the impression that many of the best position players in the major leagues did in fact pitch in Little League or through High School, or even through college. Babe Ruth is the obvious example of “could do it all at the major league level”, but even players like Nick Swisher on the Yankees follow this pattern. One NY area sports columnist I read pointed out (in reaction to fans howling about how ridiculous it was for Swisher to be the most effective pitcher in a blowout loss earlier this season) that Swisher did, in fact, pitch in HS, as do many American-raised major leaguers, and that throwing 75-MPH “fastballs” for strikes with just a little warming up (as opposed to a whole off-season and spring training) is pretty extraordinary with respect to the general population. In other words, yes, in another universe, Swisher COULD have been developed as a pitcher… Maybe not a MLB caliber pitcher, but the basic talent necessary is there.
Similarly, a friend of mine at work grew up playing in Little League with two current major leaguers, Jason Marquis (a pitcher) and Jody Gerut (an outfielder). This came up in conversation when I mentioned seeing Gerut hit the first HR at Citi Field at the Mets’ home opener this year. I asked him what they were like. They were, both of them, simply the BEST at EVERY POSITION all the way through HS. They could throw, pitch, hit, field, etc., years ahead of their peers. Marquis would hit bombs of HRs when 12 years old while playing with kids two years ahead of him. There was no obvious reason to him (through age 16-ish) to suggest that Marquis would end up being a pitcher (with the typically poor hitting stats for a pitcher) while Gerut would be an position player. The differences surely are real, but are only visible when their top-end talent gets refined, trained and directed at the college or minor-league level.