The 9 V a.c. supply isn’t 9 V. The power drawn by the emonTx is so small that transformer regulation of the UK “Ideal” adapter takes the voltage up to a nominal 11.6 V. It is necessary to limit the current drawn so that it causes negligible distortion to the a.c. waveform that you are measuring. At that voltage, there is sufficient current to operate the RFM 69CW radio module. If you have a mains supply that is low, or use a different adapter and you don’t have enough voltage out of your a.c. adapter, then it’s likely that the regulator inside the emonTx will fail to deliver a steady voltage, and the processor will continually reset.
The emonTx V2 always needed the 5 V d.c. supply, and the power supply inside the emonTx V3 was developed to remove the complication and additional expense of the separate adapter. Note that if you are using a Wi-fi module instead of the RFM radio, then the additional current demand will always require the 5 V USB supply. If you replace the RFM69CW with a RFM12B (which that supply was designed for), then you’ll find the emonTx operates over a much larger voltage range.