I gave the OP’s proposal more thought today. Here’s where I’ve landed.
- It’d likely lead to a modest increase in batting averages and on-base percentages.
Historically, stadiums which had big on-field foul territories were considered to be “pitchers’ parks,” because more foul balls could be caught for outs there (balls which would have made it into the stands in smaller parks). The Oakland Coliseum (home of the A’s until last year) was well-known for this; it was the last of the “multipurpose stadiums” (most of them built in the '60s and '70s), which had to have dimensions which would fit both an MLB diamond and an NFL field, and that was likely part of why foul territory was so big there.
I looked for information on how many foul outs occur per MLB game, and didn’t find much; Google AI indicated that, in 2022, an average 9-inning MLB game had 1.43 foul outs (but didn’t give me a solid cite for that). Assuming that that number is more-or-less accurate, it means that, on average, an MLB team would have 0.7 foul outs “back” per game – that is, what would have been an out simply becomes a strike, because it reaches the stands. Of course, that doesn’t directly translate to 0.7 more hits per game; it just means that the batter gets one or more additional bites at the apple, to potentially make that at-bat into a hit or a walk.
- It probably wouldn’t have a big impact on hitting strategy.
Most modern MLB hitters already follow a strategy of trying to hit the ball in the air, and with power (and, obviously, trying to hit it in fair territory), in hopes of a home run, an extra-base hit, or (if there are runners on base) at least a sacrifice fly. The “contact hitter” – the guy who hits line drives and gets a ton of singles (maybe legging them into doubles) – is a dying breed.
Smaller ballparks (and the death of the big multi-purpose stadiums), the move back to natural grass, and away from the fast surfaces of first-generation Astroturf carpets, and advanced sabermetric analysis of every hitter and every at-bat, have all pushed towards fly balls as being the optimal hitting style. And I’m not sure that the lack of foul territory changes that calculus much, if at all.
- Wild pitches and passed balls would become less troublesome.
Those two events (the former ruled the fault of the pitcher, the latter ruled the fault of the catcher) can have the same net effect: a pitched ball gets away from the catcher, typically rolling or bouncing towards the backstop; if the ball is loose for long enough, runners can and do advance.
As @Railer13 notes, with minimal foul territory behind home plate, wild pitches and passed balls can’t get as far away from the catcher (or the pitcher, for that matter), and thus, they’d be less likely to give runners an opportunity to advance.
- Injuries – among players and potentially fans – would likely increase somewhat.
I already touched on this above. With little to no foul territory, fielders may be more likely to run into the fence (or fall into the stands) while chasing fly balls, players and coaches in the dugouts would have less time to react to react to a foul ball (or a loose bat) flying towards them, and fans might be more likely to be struck by foul balls. (And, as @running_coach notes, runners rounding third base might also run into trouble with the wall.)
More netting/fencing could likely help protect fans and dugout occupants, but probably wouldn’t help the fielders (or the runner rounding third).