The informal statement is not true, as shown by the following programming language. Any string of, say, ASCII characters is a valid program and the meaning of every program is, "Output a program that ignores its input and outputs a copy of itself." Thus, every program in this language is a compiler for the language but the language is not Turing-complete.
I'm not sure if your "computability theory version" is equivalent but it is also not true. As I recall, for any coding of Turing machines, there is a TM that accepts its own coding and rejects all others.1 This machine is a counterexample to the proposition. If my memory is faulty, we can achieve the result by choosing the right coding. For example, let every odd number code the machine $M$ defined by "If my input is odd, accept it; otherwise, reject" and let the number $2x$ code the machine coded by $x$ in your own favourite coding scheme for Turing machines. $\langle M\rangle$ is in the language $L$ accepted by $M$ but $F_L$ is not Turing complete.
1 I think this result was mentioned here or on tcs.se recently but I can't find it, now. Can somebody remind me what it is?