Say a pitcher has a strong game, either at home or on the road. Say he reaches his pitch count at around the time an inning ends. The manager has to take him out. (Assuming there's no perfect game or no-hitter on the line. Otherwise he stays in.) It's better that a pitcher gets to walk to the dugout, both for fans, who get to shower the pitcher with respect, and for the pitcher who gets the opportunity to doff his cap if it was particularly good. Or touch the brim at least to acknowledge and accept the respect and recognize that the effort was good, if not great.
It feels cool to give a pitcher a standing clap, and it must feel cool for the pitcher to receive it. Neither happens when the pitcher gets the handshake while on the bench.
How will managers do this? They could send the pitcher for one batter, which would backfire if the batter gets on base. That becomes the pitcher's runner, after all. You couldn't have him run out and not warm up, because that would be a short jaunt. If the pitcher does warmup, that goes against his "pitch count" (not the official one, but the amount of pitches he makes during a day). There would also need to be a signal to the fans and media covering the game that he's warming up just to get the walk off. Otherwise they might think there's something wrong. It also wastes time: the relieving pitcher is entitled to have his on-the-mound warmup pitches, and he'll no doubt take them. But there's still gotta be a way to get the pitcher his show of respect from the fans.