You are correct about battery usage. The analysis they published[1] mentions this (in the 2nd paragraph of the introduction) :
"One difficulty of using Bluetooth in phones is that the operating system does not allow apps running in the background to use the Bluetooth advertising system. Hence, to do so,either the app must stay in the foreground, which drains the battery a lot, or the operating system must be changed. Apple and Google allied to provide a standard Bluetooth API which would not drain the battery. Hence, automated contact tracing apps must either be complian twith this API or make the user upset about heavy battery usage."
Singapore's TraceTogether app doesn't use GAEN api and does have this issue [2].
France's StopCovid app uses a different protocol called ROBERT[3], and they asked Apple to allow their app to run in the background[4]. So they also have battery issues.