Electronic – ac/dc adapter input amperage rating

You don't want your whole house electrical system arbitrarily floating. First, just a tiny leakage or static electricity could charge it up to high voltage. The whole system would have enough capacitance to ground so that the discharge could cause damage.

Then what if any of the three lines were accidentally shorted to ground? You wouldn't know anything happened if one of them got shorted, but suddenly other parts of the system are at lethal voltages.

Here in the US, there is a final transformer near your house, often on a utility pole at the street in front of your house. That makes center tapped 220V from whatever the higher voltage feed on the utility pole is. These are left isolated on the pole and all three lines brought into the house. The center tap is then grounded with a thick cable to a copper pipe that goes into the ground, or to a ground stake just for that purpose. This leaves two phases of 110V with opposite polarity. Most circuits are connected between one of the phases and the center, called the "neutral". A few special high power circuits, like for a clothes dryer or electric range, are connected accross both ends and are therefore 220V instead of the usual 110V.

Part of the reason for the ground setup it to deal with lightening as best as possible. The system is grounded as closely as possible to the breaker panel in the house to minimize the common voltage on the system due to ground offset. Imagine what would happen if the transformer secondary center tap were grounded at the utility pole instead of your house, and there was a nearby lightening strike. There could easily be multiple kV offset between where the center tap ground and the actual ground potential other things in your house might be connected to, like the concrete floor you're standing on in the basement, water pipes, etc. Even well insulated and properly designed appliances aren't going to protect you from that.

That's the correct indication for the transformer in its original configuration.

With a normal outlet, the neutral and the ground are bonded together at the service entrance, so the tester will light two lights: one between line and neutral and the other between line and ground, since the same voltage appears across both pairings.

With the tester plugged into the transformer, there's voltage between line and neutral created by the transformer secondary, but now there's no path between your isolated neutral and ground, so now that light on the tester no longer lights, indicating "open ground".