Thanks everyone for your help, much appreciated. Now have a happy daughter!
It was a voltage issue. 4xAAA (non-rechargeable) was 6V and just on the 6-12V it said it needed, but still not enough. And I guess the old 7.2V NiCd battery wasn't really giving out 7.2V, at least not when loaded.
Didn't chase Amazon for a manual, but those links were really helpful, thank you. I think it is the same model or very similar. Also helps me figure how to control it now it powers up the motor.
Eventually I tried 6xAA batteries and it worked! Will now look at getting a more suitable battery.
"A 320A ESC for $20." - I don't think it's a very high quality one :-)