I didn't test it, but you are on the right path. Triggered Player Starts do change the Enabled setting.*
Make a Trigger, and change it to a TT_PlayerProxy (I think) and set the trigger event to the tag you set on your Triggered Player Starts. Then you want to find the setting in the Trigger bTriggerOnce, and set it to True. (So when you pass over it twice it wont reactivate the last spawn point, and shut off the new ones)*
In Single Player mapping for games like Star Wars Republic Commando you use the Triggered Player Start for checkpoints in the map, I have a tutorial for setting them up for Republic Commando if you need help it may be of use,
here it is. The main difference is that you will want to be triggering 6 player starts at each location, just give them all the same tag and that should do it.*