Just because you are traveling with an A List member doesn't mean you get to board with them - even if they are kids. If your kids are 5 or under, SWA will allow you to board after the "A" group is finished. If this guy is an "A List" member, he knows this rule and was just trying to get on first with his kids rather than waiting to board with them later and, possibly, not be able to sit together.
I tried this once and learned that there is no negotiation. It is what it is. Instead of arguing with the gate agent, I now get on the plane first and save seats for my wife and kids who usually board with the "B" group. It's a screwy system but it's a lot better than when it was first come first served. People would get to the airport two hours ahead of their flight to board first. Meanwhile, I'm running to the gate at the last minute and end up in a middle seat, in the screaming infant section, even though I was flying more than 80 segments a year with them.
All the airlines have different boarding rules. It’s hard to keep up. One of them just changed their sequence. They board disabled and then families. It’s a disaster. The kids go running down the jet way endangering the little old ladies with canes. The wheelchairs can’t get out of the jetway because the families have left their strollers and carriers all over the place by the door. The families put their luggage in first (even though they’re in coach) and those in the bulkhead have to put their luggage way in the back. What are they thinking??