Later this week we're flying AA SEA-LAX-SYD in J. I had the option of booking it all on one day, or split it up SEA-LAX, 36 hour layover, LAX-SYD. Since the connection time is greater than 24 hours, it becomes a layover. I booked my ticket on AS for 85K miles. For my wife and stepchild I booked the same flights using 65K AA miles per seat (I had only enough AA miles for two tickets, otherwise I would have put all three on AA miles). The difference, AA wouldn't allow the 36 hour layover so even though I could do a 36 hour layover with the AS ticket it wasn't taken because we want to travel together. So is a 36 hour layover worth an additional 20K? Nope.

Now how do you actualize this. Find the availability, then do a multi-city search with the desired dates. The only thing you might miss are flights where the segments are married. For my SEA-LAX-SYD flights, the segments are married. When I searched for just LAX-SYD or SEA-LAX then 36 hour layover then LAX-SYD, the AA flights didn't show. The one thing you can try if you come across the married segments issue, and it's very risky if the layover is necessary, is book the itinerary without the layover then go back and attempt to change one of the flights to get the layover. I did the search on this after I did the original booking and the AS website actually allowed it - though I didn't accept.

Is it a true benefit? YMMV. It depends. If it comes with a significant rate increase, then not so worth it.