Consider the following argument against quantifying over everything.
"It can't be possible to quantify over everything, because if you did, there would have to be a set, your domain of quantification, which contained all objects as elements. However, this set would have to have to contain all the sets. But there can be no set of all sets, by Russell's paradox argument."
I claim it's unsound for the following reason:
We presumably can quantify over all the sets (e.g. when stating the axioms of set theory). So, if (as this argument assumes) quantifying over some objects required the existence of a set containing all the objects quantified over, we would already have a set containing all the sets, hence Russell's paradox and contradiction.
Thus, meaningfully making an assertion about all objects of a certain kind does NOT require that there's a set containing exactly these objects.
BONUS RANT: Why would one even think that where there is quantification there must be a set that's the domain of quantification? Because of getting over-excited about model theory I bet. [warning: wildly programmatic + underdeveloped claims to follow]
Model theory is just a branch of mathematics which studies systematic patterns relating what mathematical objects exist and and what statements are always/never true. It's not some kind of Tractarian voo-doo that `explains how it's possible for us to make claims about the world'. Nor do sets (e.g. countermodels) somehow actively pitch in and prevent claims like "Every dog has a bone" from expressing necessary truths!