One of the big appeals of ARM-core processors are their low power consumptions.
I'm not concerned about power usage, but about the necessity to supply power to the device, which requires dedicated hardware. Power supplies nowadays are indeed quite small, but one of them would still probably be bigger than the device itself.
From my experience, once a project like this is complete and working, people find all sorts of applications.
For example, a lower-power, small and inexpensive bitcoin node could be integrated in certain point-of-sales hardware, or it could come with bitcoin miners and run P2P pool by default (helping to decentralize mining efforts).
This hypotetical device and a POS have completely different (and actually opposite) design goals: a POS must act as a wallet but doesn't require full node capabilities, while this project wants to create a full node without wallet functions.
Mining is a completely moot point. Given the current difficulty, even a million ARM processors wouldn't even be noticed by the network. They would only waste power.