Your sketch of a proof does not appear substantially different from the proof given in section 3.1 of the linked paper. Section 3.2 explains that this result does not apply to strongly normalising languages.

See also the slides titled “So what about, e.g. Fω instead of ℕ ⇒ ℕ?” in Gergo Erdi’s talk on the subject of the linked paper:

https://gergo.erdi.hu/talks/2015-11-fomega/FOmegaUnquote.pdf

Note that your sketch is one particular variation of a folklore argument, best recalled by Conor McBride in 2003:

https://mail.haskell.org/pipermail/haskell-cafe/2003-May/004...

Here’s what Conor McBride has to say on the subject in 2015:

https://www.reddit.com/r/haskell/comments/38wels/conor_mcbri...

If you are interested in discussing this further, please feel free to contact me privately.