6 is the nominal number for a 20A circuit. That leaves a bit of a buffer.

If you want to control multiple receptacles on different circuits from a single switch, you should look into X10.

X10 is expensive as compared to just a switch or switches. You could also do multiple wall switches, one for each circuit. If you are dead set on one switch to control them all, you can go with a multiple pole electrically held contactor with 20A rated contacts and 120V coil. The switch energizes the coil and each circuit runs through each contactor contact.

I just have standard always hot circuits. I just turn my games on one at a time from the game itself. If I want to play one I don't want to turn on a bank of 6 to do so.

Wrong voltage on the magnet to compare to the lines voltage.

Actually a DMD games does draw 2.5A. Two of us measured with a Fluke 87 True RMS meter and 1000 clamp on ammeter in both attract mode and playing set on the recording setting. The mains 8A fuse is not sized based on the actual load of the game, it is sized to protect the WIRING from short circuit conditions first and then overload conditions second.

That's not the issue we are discussing! Here, I copied what I responded to below:

You are correct that 4A is 4A; but what he is stating is that if the 50V line has a 4A fuse then you need at least that for the main fuse because it has other coils and lights so the entire game is going to pull way more than 4A.

4A x 50V = 200W. Across the transformer --> 200W/120V = 1.6A. The 4A on the 50V secondary only translates to 1.6A on the 120V line side. Amps are amps, but not when talking about different voltages on different sides of the transformer. Wattage on the other hand is the same.