Is the behavior you're seeing consistent with a noise gate? A noise gate is hardware or software that silences an audio channel whenever the current signal level falls below a preset threshold.
IIRC, the meetinghouse PA systems have some noise gating built-in. However, there might be other noise gates in the Zoom software, whatever analog-to-USB converter you might be using, the laptop's operating system, etc.
One somewhat easy way comes to mind to verify whether or not a noise gate built in to the PA system might be the cause of the problem you're seeing: 1) Tap off the analog signal between the PA system and the laptop with something that can feed a pair of headphones with fairly good sound isolation from the external environment, preferably with a lot of gain that the headphone wearer can control easily; 2) Have someone with a good ear listen to the headphones while someone speaking goes between right on the microphone vs. off in the distance. If the headphones go silent at the same times as the Zoom broadcast goes silent, then that would be evidence that the PA system is noise-gating the analog signal before it ever gets to the laptop. If not, that would be evidence that the noise gating is downstream from where you tapped off the signal.
The newer software-defined PA systems could have noise gates and other signal shaping behaviors built in to various paths through the system. If it turns out the signal you're currently getting from the PA system is noise gated in a problematic way, one solution might be to tap off a speaker-level signal from the overflow channel and attenuate it down before feeding it to the laptop. Another solution might be to set up a "house" microphone and mix that in with the signal from the PA system, switching between the two as desired.