Once upon a time Medium/high level ATC actually was slightly more compartmentalised than it is now, back in the days when skies were less busy and aircraft were slower, but that doesn't work well anymore, but I'll come back to that.
I'm not sure what point the TV report was trying to make but Liverpool isn't special, all airports have their own ATC system at the (literally) local level to handle the controlling of aircraft on the ground and (usually) within a few thousand feet/handful of miles of the airport, where high speeds aren't involved and distances don't get covered quickly...those controllers manage the local box of air..
Problem is go beyond the local area and as aircraft get higher up speeds usually increase, ground get covered more quickly and a more centralised system makes more sense with controllers looking after large sectors of airspace, not little local boxes.
Crossing England at high level a pilot might only need to switch controllers and radio frequencies half a dozen times...which reduces everyone's workload, improves safety, certainly helps with coordination.
It also helps if everyone on the ground side is in the same building...even going way back ,1960s, England and Wales only used to have two main ATC centres, London and Preston, but it's been London alone for a long time (that's now physically at Swanwick), Scotland has it's own centre for medium/high level traffic.
However whats being missed in some of the commentary about yesterday is that the problem wasn't one of radars not working or controllers not having comms, it was a problem that the flight planning system somehow wasn't accepting flight plans for flights originating out of some of the southern airports...and flight plan handling systems by their nature can't be too compartmented, data has to be shared across ATC centres and airports.
What is also maybe important to note is that London ATC was still happily handling those flights they did have flight plans for e.g. overflights of England and Wales, and those flights inbound to southern UK airports (until parking space ran out.).