You can probably beat a true RTW fare by buying point-to-point, but it also might depend on the age of the kids. Kids under 12 receive a 25% discount on Oneworld Explorer RTW tickets - go to http://www.oneworld.com and follow the links to the online booking engine.
Alliance-based RTWs are priced differently depending on where they're purchased. At present the UK is much cheaper as an origin/destination point than the USA, so if this is an annual thing, it would be good to start the "cycle" in the UK. RTW tickets are good for a year.
The base price for a 3-continent/16-flight Oneworld Explorer sold and started in the UK is around US$2580 plus taxes and fees. A quick back-of-the-envelope investigation of LHR-ICN-TYO-LAX-LHR using one-way fares came pretty close to that, but fares are quite dependent on the dates traveled so not knowing your dates this is a stab in the dark. Most likely the "taxes and fees" would add another several hundred dollars to the total for the RTW, but that price will be fixed for a year, whereas the point-to-points can fluctuate all over the place.
The RTW's real advantage in this scenario is that 16 flights are included, of which up to six can be within North America. So assuming the Korea and Japan stops are both to be done in rapid succession, with no time for additional touring in Asia, but also assuming that your "stopover" in LA is for a longer period of time, before you "return" to the UK, then you could use the North America flight allowance for any other personal or work-related travel in the interim.
For example, say you fly from Japan to LA to go home. But before you return later in the year to the UK, you could use the North America allotment to travel to, say, Alaska, or the Caribbean, or Central America, or New York, or... you get it. ("North America" includes the US, Mexico, Canada, the Caribbean and all of Central America.)
So in that sense, the additional initial cost of the RTW has now potentially turned into quite a bargain. Instead of four flights for $2600+ or an average of $650+ per flight, you've gotten, say, 10 flights for the same price, including 3 intercontinental flights at $260 each, so much better leverage. You can amend the itinerary in an RTW for a small change fee ($125 in most cases) and change dates at no charge, so they're not as "inflexible" as many people feel.
It really comes down to your personal travel patterns. RTWs can be cost-inefficient, or they can be spectacularly cost-efficient, depending on how they're used.