I can disable all plugins (besides Broadcast) and still get the Not broadcasting error. I have cloned a couple of test sites to broadcast to and on the post listing page I can Restore / Find unlinked children and the posts already present when the site was cloned will show that they are broadcasted to those clone sites.
PIggybacking on this since OP is a former coworker who I inherited this project from 🙂
If Ajax Search Pro is not enabled, the WP_Broadcast: No _POST available error is shown. If I enable it, the error returns to WP_Broadcast: No _POST available. Not broadcasting.
As mentioned previously this is indeed (sadly) hosted on WPEngine but will look at other options if WPE is typically an issue with this plugin.