Converting final state PDA to empty stack PDA
Your description of the construction is wrong. The new automaton has a new initial stack symbol, from the new start state it places the original PDA's initial stack on top of it and moves to the original's start state. The new initial stack can only be removed in a new state that is entered via $\epsilon$ from final states, and in this new state all that is done is to slurp down the stack's contents.

This is activated from any final state, and the automaton can end up with empty input and empty stack only if the original reached the final state at the end of the input.