I know how to do this.
It does involve teleporting but instead of disabling and enabling them, just use a trigger_changetarget to change the teleport destination and keep it simple.
For the "rounds" system, use game_counters. Each counter has the same name. For example if you have 3 spawn sets, you would need a setup like this.
3 counters of the same name gets fired at the end of a round. ex: "roundcount"
A counter fires its target at 1, 2, and 3 (that's why you need 3) you start on spawnset 1 with the counter initially at 0. So when it fires roundcount, round 2 starts at 1 and also fires the changetarget for spawnset 2. When it gets to 3, reset to round 1 settings.
I think that's it... I can perhaps make a template, I need to update my KOTH setup anyway.