Yes, Chuck, you're right. In the UK it is not lawful to conduct flight training for ab-initio PPL training ftom anything except a licensed or government aerodrome.
To 'license' an aerdrome is an expensive undertaking and the aerodrome operator has a host of CAA requirements to meet. So what some PPL organisations do is to operate from an unlicensed aerodrome, perform a toch-and-go at a convenient nearby licensed aerodrome, do the lesson, than the FI lands at the unlicensed aerodrome. The lesson is considered as the period from the touch-and-go at the licensed aerodrome (for which there will also be a landing fee - remember, this is the UK!) until the FI takes over for the landing..... It's a fiddle to get round the law - but it's a very stupid law.
The idea is to protect people from the unscrupulous operator who would operate from a muddy cow pasture with a shabby old spamcan and an ancient caravan (if that). But what is being sought now is a more commonsense approach with less black-and-white distinction between licensed and unlicensed aerodromes. Not a free-for-all, but approval to conduct particular forms of training from 'approved training sites' depending on their individual standards. But it isn't going to be an overnight change.
And don't forget that, whereas in the US an aerodrome is seen as a normal facility supported by local commerce and part of the national transport infrastructure, in the UK they're usually viewed as an annoyance to vociferous local noise campaigners - and such little aerodromes NEVER receive funding from local chambers of commerce or other supporting businesses.
US - Land of the Free
UK - Land of the fee