Using Rice's theorem to prove undecidability of $E_{TM}$
You have to be careful when you talk about accepting empty strings. Using the language from the linked question, what you are suggesting would yield $L(M_x)$ = $\{\epsilon\}$ instead of $L(M_x)$ = $\emptyset$.
You can use the same construction for machine $M_x$, but just change the the non-trivial property to &quot;whether this machine accepts on no inputs&quot;. It's clearly a non-trivial property since it's not empty (there are machines with empty language), and it's not the set of all Turing Machines (since not all Turing Machines have empty language).