I don’t even thing you should limit it to “designated home teams pick from designated visitor teams”.
In the NFL, if Tampa Bay (2 seed) thinks that they have the best shot against Dallas (3 seed) due to an injury or just general matchups, they should be able to make the Cowboys go on the road. Sorry Jerry, you can’t host this week. Should have won more games.
I don’t think baseball gets much more interesting, unless the #1 team wants to dodge a potentially hot and streaking team coming out of the wildcard (even if they just threw their best guy).
NHL probably works with 16 teams in. NBA does too (though add the wrinkle of having to pick before the play-in games are done). Honestly, the first round needs more spice. In the NBA, you might also try this with the push that’s been made to stop allotting berths by conferences. I’m not sure the NHL would be interested in getting rid of the traditional trophies for semifinalists, though they did modify it to not be conference specific due to COVID last year.
I don’t think it works in something like March Madness. The field is too big, the sites are too diverse and games are too close together. Also, filling out the brackets is a big part of the fun. I also don’t think it makes in knockout soccer competitions where teams are randomly drawn.
Most teams will take the normal seeded matchups, but the times when that doesn’t happen will be intriguing and full of bulletin board material.