I've just set Strictly to series record from series 23 and onwards without problems.
Have you used the /check-services/ function to check for problems in your area? You can also a run a test on your equipment from there. If nothing is showing you could also try the automated Service Status number 0800 561 0061. This is generally better as it gives details of more local issues down to postcode level.
You say that you've turned off your TV and box off at the plug.. do you mean your TV360 box and actual TV, or TV360 box and hub? If you haven't turned off the hub then try rebooting that, although to be honest I can't see that making any difference.
Andi if that doesn't help then try a factory reset.
To factory reset go to Settings (the cog at the end of the Home menu) > System > Factory Reset. It's actually misnamed and only a partial system reset and not a complete factory reset. You are given 2 options
Keep Recordings
Format Disk
Select Keep Recordings, which means you won't lose your recordings, and any planned recordings, or series links you've set up. You may have to sign into your apps such as iPlayer, Prime, Netflix and ITVX again.