How to prove that ε-loops are not necessary in PDAs?
The removal of $\varepsilon$-transitions has been studied for the more
general model of valence automata by Zetzsche [1]. 
Valence automata are essentially finite automata with a monoid for storage.

Among other things, Zetzsche shows that for monoids $M$ from a rich class $\mathcal{C}$ of monoids
(which contains (partially) blind counters, stacks, and combinations thereof), 
$\varepsilon$-free $M$-automata accept the same class of languages as $M$-automata.

Since PDAs with a $k$-symbol stack alphabet correspond to valence automata over
monoid $\mathbb{B}^{(k)} \in \mathcal{C}$ ($\mathbb{B}$ is the 
bicyclic monoid),
the result from Theorem 1 (resp. 7.1 in the preprint) applies here. 

The proof is long and technical; the proofs of lemmas 8 and 10 (resp. 7.6 and 7.9) 
contain the relevant constructions. Note that while they do not use grammar models
(as required in the question) they do use valence transducers.




Silent Transitions in Automata with Storage by G. Zetzsche (2013) [more elaborate preprint on arXiv]