I like the idea of using a scheduled task, but your code has several problems with it. I'm frankly stunned that your code works at all. The issues:

* DynamicModule must display in order for it to function. Your DynamicModule displays no result.
* DialogInput freezes all kernel activity until it gets an answer. There is never any reason to use it unless the output of your Shift+Enter evaluation directly relies on an input from the user, and having it interact with Dynamic functionality is very, very dangerous.
* DynamicModule does not, by default, cross through notebooks. You need a DynamicModule wormhole to do that (although my solution does not do this because I contain the scope of the DynamicModule better).
* DynamicModule variables can't be accessed from scheduled tasks. Sorry, just a limitation of the system.
* You used a variant of RunScheduledTask which only runs the task once (by wrapping curly braces around the second argument).

Here's how I would do this:
Note that I use CreateDialog here so that the Shift+Enter evaluation returns immediately. I use TaggingRules on the dialog notebook in place of a DynamicModule variable (this is a well-documented meme...see the documentation for TaggingRules). I use With[] to pass EvaluationNotebook[] into RunScheduledTask[] because EvaluationNotebook[] would be undefined in a scheduled task. I use a DynamicModule in the dialog notebook to fire off an initialization, but it doesn't track any variables.

Thanks Vitaliy,
I can imagine that the goal appears to be obscure. I'll expand on what that goal is, and how it relates to teaching with Mathematica,  below.  Perhaps the explanation of the goal should be in a new community discussion, as it is off-topic from this original question. 

Thanks John,
I'm stunned when my FrontEnd codes work as well!  I am embarrassed to admit how much time I put into making this work---more embarrassing is that it is several generations removed from one of your solutions to a related problem.  The fruits of too much inbreeding, and letting children (i.e, amature programmers like me) play with sharp objects.  I would never have thought of combining CurrentValue, Tagging, and RunScheduled task.  However, shouldn't the scheduled task be removed when finished?  Also, if I have many such dialogs, don't I need to restrict the scope of "count" to each cell? This scoping is what led me down the path to the DynamicModule.

 I'll adopt John's solution to put up signposts (think Burma shave).  I'll also need to adapt John's previous solution to this pause technique for material that contains evaluatable expressions.  John, if you have read this far and would be kind enough to help me out I would be very grateful---I'll send you the code I have for this so far.  I fear that my attempts to develop the vehicle to teach are preventing me from developing my materials science content as quickly as I would hope.