Are festivals taking too long because of all the changeovers?

Yes — if the gaps are five minutes, the six-minute game is not really solving the problem. I would make the rotation almost automatic. Give every team a colour, every pitch a number and decide the movement before the first game. For example, one colour stays, one moves right. Whatever system you choose, keep it simple enough that an adult can explain it in 20 seconds. Have the next groups standing beside the pitch before the current game finishes. A central timekeeper helps as well. One clear start and finish signal for all pitches is much quicker than every facilitator running their own clock. Most delays come from adults reorganising things the children did not need reorganised. Short games work when the next game starts quickly.