I’m not familiar with binary lambda calculus; it’s been over 35 years since I thought extensively about reflective languages. But note even in a language like Haskell the lazy semantics comes from the interpreter itself. The proposition that no language (mathematics) can represent all propositions goes back to Russel/Gödel.

Perhaps the better answer is at the end of my comment: for all intents and purposes you can get essentially all the value of a reflexive language through other means, even if those means are much less elegant.