I'm not an electrical engineer or an electrician, but I think the reason is that all the 5V relay does is provide current. It doesn't close the circuit, which is what you're looking to do.
The relay, when powered by the 5V adapter actually closes the circuit, which is what you need to make the fireplace work. So let's look at it this way: If I cut the wires from the wall switch, and touch the negative and positive leads of a 5v battery to each of them nothing will happen. But if I cut the wires from the switch, and just touch them together the fireplace will light, because the act of touching the wires closes the circuit to the fireplace, which already has the power it needs to light. The 5V current from the adapter is what is needed to trip the relay to automatically close the circuit.
I was never able to get it working exactly like the OP's. I am not sure whether it was because the relay he used needed something I wasn't giving it, or if I was doing something minor wrong. But I ended up buying the following items, and hooking them up. I plugged the Elgato switch into the receptacle. I plugged the AC/DC adapter into the Elgato switch. Then I ran a couple of small gauge wires from the LED light adapter tip of the AC/DC adapter to the relay, and hooked the wires that I had cut from the switch into the other side of the relay. It works like a charm.
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