It's well known that $$ \langle 0| \phi(\vec x, t) \phi(\vec y, t) |0 \rangle \neq \delta(\vec x - \vec y) . $$ It is then regularly argued that this is not a big problem since the commutator $$ \langle 0| [\phi(\vec x, t), \phi(\vec x, t)]|0 \rangle $$ vanishes for $\vec x \neq \vec y$. (See, for example, page 37 here.) This is motivated by claiming that if for two operators $O_1( \vec x, t)$, $O_2( \vec y,t)$, the commutator $ [O_1(\vec x, t), O_2( \vec y,t)]$ vanishes, "this ensures that a measurement at $\vec x$ cannot affect a measurement at $\vec y$ when $\vec x$ and $\vec y$ are not causally connected."
While this argument certainly makes sense in the context of quantum mechanics, I'm failing to see how it applies to quantum field theory. The field operators are not measurement operators in the usual sense and thus I don't see how the field commutator is related to causality.
Instead, $\langle 0| \phi(\vec x, t), \phi(\vec y, t) |0 \rangle $ is the probability amplitude that we find a field excitation that we prepared at $\vec x$ at the same moment in time at some other location $\vec y$. In other words, "particle-like" field excitations are not completely localized in QFT. This doesn't look like a big deal to me in terms of causality. (At Wikipedia it's argued that this non-localizability is a result of the unavoidable vacuum fluctuations.)
Analogously, $\langle 0| [\phi(\vec x, t), \phi(\vec x, t)]|0 \rangle $ is the probability amplitude that we find a field excitation that we prepared at $\vec x$ at the same moment in time at some other location $\vec y$ minus the probability amplitude that we find a field excitation that we prepared at $\vec y$ at the same moment in time at some other location $\vec x$. In other words, the $\vec x \to \vec y$ amplitude and the $\vec y \to \vec x$ amplitude cancel exactly. (This doesn't seem very surprising if we believe in homogeneity and isotropy of spacetime.) How is this quantity related to causality?