There are many situations where paging can cause feedback (two phones in the same group near each other, or external speakers in proximity of the paging initiator’s phone). There are external devices (expensive) that take the page call, record it, then send the page to the speakers.
Requesting there be a check-box on page groups created in the system. This check box would be “Delayed Paging” (or something to that effect), and when active, the paging initiator would hear the page tone, say their message, then hang up. The system would record the page, then page it out to the group after the call completed (eliminating any possibility of feedback).
Additionally, this option could be used if someone tries to make a page to a group that is already being accessed (the second person attempting would be recorded and played after the first page completed).
There are other asterisk-based systems that offer this (FreePBX is one of them).