[The assumption in this question is wrong. It is possible to enumerate exactly the decidable languages with semideciders.]
Lets say we have a TM $M_E$ enumerator that writes out codes of TM's on a tape. How does one show that it's not possible that every decidable language $L$ is represented, but no undecidable $L$ is represented?
If we demanded that $M_E$ write out every $M$ representing a decidable $L$, then this could be shown impossible by Rice's Theorem II (for r.e. index sets).
If we demanded that every $M$ $M_E$ writes out is a decider, then this could be shown impossible by diagonalizing out of the set.
In this setup $M_E$ writes out $M's$ that aren't necessary deciders but that represent decidable languages. Every decidable language $L$ is represented by some $M$ on the tape, and no undecidable $L$ is represented. How does one show that this is impossible?