The paper in the OP discusses this claim in section 3, and mentions that Kleene came even before that:
> We would note that Kleene seems, however, to have already had the self-referential argument earlier in his classic book from 1952, Introduction to Metamathematics
They also bring up the "symbol-printing problem" present in Turing's 1936 paper, which is trivially equivalent to the halting problem with today's hindsight. That paper has a well nuanced take with a lot of interesting information.
> We would note that Kleene seems, however, to have already had the self-referential argument earlier in his classic book from 1952, Introduction to Metamathematics
They also bring up the "symbol-printing problem" present in Turing's 1936 paper, which is trivially equivalent to the halting problem with today's hindsight. That paper has a well nuanced take with a lot of interesting information.