Although I recognize this is a matter of taste, I do not find it unnatural to fix a TM.
I find the proof by contradiction relatively concrete.  In that proof structure, the start is: Suppose there exists, for the purposes of contradiction, a TM $T_1$ that determines if any other halts.  Then we could construct a TM $T_2$ that accomplishes the impossible.  So $T_1$ cannot exist.
This is the proof in Hopcroft and Ullman's _[Formal Languages and their Relation to Automata][1]_, pp. 108-9.  

Perhaps you would find worthwhile the variant on this proof structure used in Aho and Ullman's _[Foundations of Computer Science][2]_, p. 750.
Here they do not use TMs, but rather just programs, calling $T_1$ the "decider" program, reaching the conclusion that no such decider can exist via a "complementer" program.
One can present a few wiring diagrams using the decider and complementer programs as black boxes
to make the contradiction clear.  Not-halting can be equated with infinite loops.
If your utter novices are programmers, this approach can
be more convincing than an indexed diagonal argument.


  [1]: http://dl.acm.org/citation.cfm?id=1096945
  [2]: http://infolab.stanford.edu/~ullman/focs.html