[AtlUmpSteve]

I think you are overthinking this one. Yes, I understand the ball is always live; until you make it dead!!

I do not think anyone intends the dead ball appeal to mean that, if a team doesn't request time, you cannot accept the appeal. That returns us to the cause and reason for a dead ball appeal; to simplify the process, and move on. IMO, once the ball is in the circle in the pitcher's possession, and all runners have stopped at base, such that any further baserunning would now violate the LBR, the umpires are in a position where time can be granted. If a player then makes a dead ball appeal, you can call "Time" and honor that verbal appeal. Don't go looking for the nits of a) they didn't ask for "time" first, or b) making the player repeat a perfectly clear verbal appeal after you call "time", or even c) making a major production over calling "Time". When the game conditions are right, just go with it.