All the proofs I've seen work in a similar way, by reductio ad absurdum. They assume a termination-testing program can exist, show that if it does, then a contradiction always arises, and therefore conclude the initial assumption to be false. In most cases, the contradiction is a program that stops if and only if it doesn't stop.
What Machines can and cannot do by Nievergelt and Farrer, Computing Surveys volume 4 number 2, June 1972. Available as AI photocopy N 21. Their proof has the same structure as in Strachey's letter, but the termination tester is passed the index number for a program rather than the program itself.
Incomputability by Allison and Hoare, from Computing Surveys volume 4 number 3, September 1972. Available as AI photocopy H136.