Why don't creation operators of a quantum field constantly create particles out of the vacuum? A quantum field is ubiquitous and omnipresent, as evidenced by the vacuum energy which can manifest itself only if quantum fields are everywhere, albeit being in their ground states if there are no particles present corresponding to some mode. Thus you have creation operators and a vacuum, so why doesn't it mandate that they produce particles constantly. Energy conservation is not an issue here, since the particles can always get annihilated later on, as happens for a propagator. So why don't we see particles constantly popping out and getting annihilated later on?
Creation operators are mathematical objects used to formulate quantum field theory. In quantum field theory they aren't "real things" that constantly do something. Similarly a propagator isn't something that gets annihilated.
On the other hand the picture of particle-antiparticle pairs being constantly created out of the vacuum is a common one in quantum field theory. This picture is based on the idea that the quantum states where we count particles (eigenstates of the non-interacting system) won't be eigenstates of the interacting system (even though we don't actually have a mathematical notion of states in interacting relativistic theories). Thus when looking at the system with that basis in mind one would expect to see "quantum fluctuations". The picture of particles being constantly created (and annihilated) in vacuum isn't bad.