Also, any change in date, ticketed points (origin, destination, or a stopover point), or number of flights requires reissuance. (Implicitly in your statement, changing the operating carrier also requires reissuance).

The problem isn’t actually that the tickets don’t get reissued — that’s normally not a problem. They’ll be queued and then reticketed. The problem is normally that the ticketing carrier doesn’t always push though the updates to the operating carrier. Thus, the operating carrier sees old ticket numbers, which will have a status of “Exchanged / Reissued.” As far as they know, the ticket could have been used on some other flight on another carrier. Therefore, the flight reservations will eventually be canceled for lack of a valid ticket. So, it’s less important to insist upon reissuance than it is to ensure that, when the ticket is reissued, the ticket numbers change on your United receipt and the new number is reflected on the operating carrier’s reservation.