You might not find the term "speed up rule" in the rule book, but speeding up the game was the original intent of the courtesy runner rule, which is why it is limited to the pitcher and catcher.
At the end of a half-inning, the catcher needs a little extra time to put on her gear, plus you want the pitcher ready to start taking her warm-up throws. Both are delayed if F1 or F2 are on base when the third out is made. So the rule doesn't really speed up the game as it's being played, it speeds up the downtime between innings.
Some organizations, or local leagues, have since expanded the use of courtesy runners beyond that original intent. They might use the rule to promote participation (get more players into the game), to allow for an injured runner to be replaced or even for coaches to use this as a strategic tool to get a faster runner on base.
And I'm not really sure that the original rule does all that much to speed up the game anyway. I've seen plenty of typical courtesy runner situations go like this:
Coach wants a courtesy runner, but is unsure of the rule so he has to ask the umpire which player can run, they have to take time to sort that out first, then they have to find a player in the dugout who fits the bill, then once they find her she has to find a helmet, then she has to find a helmet that fits...and then the catcher still isn't geared-up when the half-inning ends and they still lollygag around before starting their warm-up pitches.
At the end of a half-inning, the catcher needs a little extra time to put on her gear, plus you want the pitcher ready to start taking her warm-up throws. Both are delayed if F1 or F2 are on base when the third out is made. So the rule doesn't really speed up the game as it's being played, it speeds up the downtime between innings.
Some organizations, or local leagues, have since expanded the use of courtesy runners beyond that original intent. They might use the rule to promote participation (get more players into the game), to allow for an injured runner to be replaced or even for coaches to use this as a strategic tool to get a faster runner on base.
And I'm not really sure that the original rule does all that much to speed up the game anyway. I've seen plenty of typical courtesy runner situations go like this:
Coach wants a courtesy runner, but is unsure of the rule so he has to ask the umpire which player can run, they have to take time to sort that out first, then they have to find a player in the dugout who fits the bill, then once they find her she has to find a helmet, then she has to find a helmet that fits...and then the catcher still isn't geared-up when the half-inning ends and they still lollygag around before starting their warm-up pitches.