Impredicativity greatly increases the logical strength of a formal system, and impredicative propositions are also a consequence of various axioms including LEM and Zorn's Lemma. An impredicative sort of propositions is built-in to CiC. A propositional resizing axiom may also be included in type theories for its own sake.
On the other hand, Andrej Bauer wrote that "those who feel uneasy about impredicativity, and those who want to carefully callibrate the logical strength of their formal systems, look for ways of avoiding impredicativity [...]"; Lawrence Paulson has "personally heard Martin-Löf criticise CIC, in particular because it is impredicative"; and unlike Coq, Agda's sort of propositions is predicative.
The advantages of adopting impredicative propositions seem clear, but despite the apparently widespread concern, it's not clear to me what the trade-off is. For example, classical axioms are powerful, but come at the cost of breaking properties like canonicity and strong normalization; and Cubical TT is powerful, but is a trade-off against alternative systems like Setoid TT.
What is the trade-off to accepting impredicative propositions?