I think there are two issues to consider.
Be concrete
Sequence diagrams are at their best when they are used to convey to a single concrete scenario (of a use case for example).
When you use them to depict more than one scenario, usually to show what happens in every possible path through a use case, they get complicated very quickly.
Since source code is just like a use case in this regard (i.e. a general description instead of a specific one), sequence diagrams aren't a good fit. Imagine expanding x levels of the call graph of some method and showing all that information on a single diagram, including all if & loop conditions..
That's why 'capturing the essence' as you put it, is so important.
Ideally a sequence diagram fits on a single A4/Letter page, anything larger makes the diagram unwieldy. Perhaps as a rule of thumb, limit the number of objects to 6-10 and the number of calls to 10-25.
Focus on communication
Sequence diagrams are meant to highlight communication, not internal processing.
They're very expressive when it comes to specifying the communication that happens (involved parties, asynchronous, synchronous, immediate, delayed, signal, call, etc.) but not when it comes to internal processing (only actions really)
Also, although you can use variables it's far from perfect. The objects at the top are, well, objects. You could consider them as variables (i.e. use their names as variables) but it just isn't very convenient.
For example, try depicting the traversal of a linked list where you need to keep tabs on an element and its predecessor with a sequence diagram. You could use two 'variable' objects called 'current' and 'previous' and add the necessary actions to make current=current.next and previous=current but the result is just awkward.