Most of the lower-tier fares on most carriers will have "no stopover" rules, though you could still stay up to 23:59 between flights without it counting as a stopover (vs just 4 hours domestically).

I tend to really max out stopover and open-jaw rules on AWARD tickets, where things tend to be more flexible. Once did a 25K award out on MNL on CO that went MNL-GUM (daylong non-stop stop)-MAJ (final destination, stop)-YAP (stopover)-GUM-MNL. Why see just 1 place when you can see 2 or 3?

Also once did ATL-ROR (stop)-MNL (stop, open jaw), BKK-ATL even though the agent swore ROR wasn't an allowed stop. The online booking took it.

I guess the legacy airlines figure that if people did stopovers on paid tickets, anarchy would result. They're only slowly getting comfortable with the idea of one-way travel. Change takes time.