In logic, why do we talk of universal closure of a formula, and don't consider its "existential closure" (as far as I know)?
I guess that one of the reasons may be that interesting systems like PA are written down much more succinctly and naturally without the universal quantifiers. I think there might be more to it than that, though.
If we want to talk about existentially quantified things, we often want to refer to them by name, so it's usually a good idea to name them by Skolemisation: instead of $∃x.P(x)$ introduce a constant symbol $c$ and write $P(c)$; instead of $∀x.∃y.P(x,y)$ introduce a function symbol $f$ and write $∀x.P(x,f(x))$. This limits the number of situations in ordinary mathematics where taking the existential closure would come in handy.
[This entire answer was written by user Z. A. K.]