Water – Will keeping the hot water running by accident cause any damage to the hot water heater

We have a circulation pump that was in the house when we bought it. The instant hot water is nice, but it is a huge energy waster, as you suggested. Our solution was to put switches on the circulation pump. It's plugged into a receptacle near the water heater, so it was as easy as making that a switched receptacle. We have switches (timer type that turn themselves off soon after you turn them on) next to the shower and by the kitchen sink. It only takes about 30 seconds for the circulation pump to get the hot water to you, so we just turn it on while getting ready for the shower (or while stacking the dishes). So convenient, but without the wasted energy or water.

The final piece of the puzzle reveals what was going on.

What I thought was a 2nd or 1st building shutoff was a shutoff outside the building that only shuts off those two specific bathroom sinks' cold water supplies.

Why!? Who knows! It's an old building.

But that explains why those two taps were too hot and why none of the other fixtures in the building were affected.

Another factor was that I was using the kitchen sink faucet to test the water temperature, but those are probably not anti scald.