Entanglement entropy with a local operator insertion

According to several papers discussing the behavior of entanglement entropy (EE) with a local quench or a local operator insertion (cf. https://arxiv.org/pdf/1911.04797.pdf), if the operator was inserted far away from the subregion, it does not instantly affect to the reduced density matrix and hence EE is same as the vacuum EE. (This is the statement.)

I think this is due to causality; the domain of dependence of the subregion does not contain the point of the operator insertion before some sufficient time.

However, I think this explanation is puzzling since EE of any pure state should equal to EE of the complement and the domain of dependence of the complement region should include the point of the operator insertion. Furthermore, in case the local operator is not unitary, I think the reduced density matrix cannot equal to the one without the operator excitation. (This is my question.)

Is there any simple, intuitive explanation for this seemingly contradicting statement?