You should be able to test OBS without making noise in your space: on your mixer, turn down the "Stereo" fader in the lower right (or press the button just above the fader), and your main speakers should be silent. If Aux1 and Aux2 go to musician's monitors, turn their "SEND MASTER" knobs down to silence those monitors. (and if you are forgetful as I am, stick a large PostIt note to the mixer to remind you restore the settings when you are done...) You only need Aux3, which feeds OBS. You can play iPad, use the mics etc. with no local sound.
In your stream from April 28, there are bunch of instances where the audio volume drops abruptly for half a second or so. Your preachers are pretty passionate, so I can imagine an audio peak triggering a mis-configured compressor or other audio filter. The one-knob compressors in you mixer are set pretty conservatively, and I presume there is just a cable between the mixer and the OBS computer, so it would be good see how the OBS filters are configured.
Would you be willing to share a copy of your Scene Collection here? That would let us look at your audio filter parameters. Your YouTube credentials are in your Profile, so there shouldn't be any security issues in the Scene Collection. In OBS, select "Scene Collection", "Export." You will be prompted to save a .json file. Then use "Attach files" here to send it.
I note that the fader for mixer channel 19/20 is turned up, presumably to play audio via USB from a computer into the space (unless the channel is turned off by button - I can't tell in the photo). But the Aux3 send for channel 19/20 is turned all the way down, so any audio from that USB connection won't be sent to OBS. Probably not related to your streaming audio issues, but...