During a homework assignment in a graduate complexity course I took at Cornell back in 1985 I used the following reasoning: Since a computer code sits in RAM that a program can read, by the Church-Turing thesis we can assume a program has access to its own code.
The TA marked me wrong on the problem because I assumed the recursion theorem that we hadn't yet covered. I wasn't assuming the recursion theorem, I was assuming the Church-Turing thesis and concluding the recursion theorem.
I did deserve to lose points, the Church-Turing thesis is not a mathematical theorem, or even a mathematical statement, and not something to use in a mathematical proof. Nevertheless I still believe that if you accept the Church-Turing thesis than you have to accept the recursion theorem.
Now the recursion theorem does not have a trivial proof. So the Church-Turing thesis has real meat on it, in ways that Turing himself didn't anticipate. Since the recursion theorem does have the proof, it only adds to my faith in and importance of the Church-Turing thesis.