I love how there are so many different ways to do the one thing! There's not many bigger differences than between english and western, though. Especially english in Australia vs. Western in the US.
I always ride two handed doing any game. Because over here we do way more games, a typical gaming weekend can have anywhere from 7-15 games. There are some of these games that would be impossible to do one handed completely off leg. A lot of them are so tight that your horse needs the suppoert of a strong outside rein and weight in the buit combined with leg to get them trhough the pattern.
There are a lot of people here who use sporting reins as we call them, gaming reins as you do. I used to when I started out, but have now changed back to normal length reins. I don't like english reins, I use cotton one piece reins that are a bit longer than your average gaming reins. Here is why:
Because we ride two handed, we also turn a lot differently. My own technique uses an open inside rein to free up the shoulder and give the shoulder a pocket of unrestricted room to move into. In gaming reins, it is impossible to do this and maintain a supportive outisde rein.
Another thing is I never ride with a set rein length. Depending on my horses mood, freshness, the game, the weather, etc I will vary my length of rein. This is very difficult in gaming reins!
In between games, I let me horse stretch out and often take him for long rein trots to stretch out his back. This is impossible in gaming reins. Keep in mind we do whole weekend of gaming and are on the horses backs all day.
These are the reasons why I personally don't use gaming reins.
My mounted games coach likens games reins or knotted english reins (what many people do here as a quick fix) to trianing wheels on a bike. We are all accomplished enough riders that we should be able to hold our reins at whatever length we wish, and alter them when we wish, and relying on a knot is a step backward. But that is in regards to mounted games, which is a bit different to the gaming you guys are talking about.