I wrote down a whole spiel why this is a good idea but accidentally nuked it when i decided to log in and not post as an AC.
Abbreviated version:
I'm not an apple fanboy, but this is definitely the best solution. Keep the device locked down to only run facetime and skype, keep the carers in control of scheduling when people call and charging the ipad when it's not needed (it has a fantastic battery life anyway so overnight charging ought to suffice unless there's a busy day). Keep an applecare contract open for the device and keep the internet connection with a provider that does high levels of support (or centralised administrative group or outsourcer ) and there's 95% of the support you'll ever need. I considered the idea of a long life android tablet with a child proof launcher, but the potential for the one way charging connector was a bit of a deterrent from me suggesting that as a solution.
The roll your own box and administer it remotely/as a client is a fair idea, but requires someone, somewhere to administer the device and incurs a substantial additional cost as although it may use standards, it is a custom created monstrosity that could be a liability if things break in the future. The smart tv thing is an interesting solution, but seems like it could be a lot harder to get the device to the less mobile patients than a simple to carry device like a tablet.
the KISS principle applies here more than anywhere else it possibly could.