Why not? Old Sarum and Compton Abbas are good examples of a single grass runway unsuitable for CAT but licensed and doing PPL training
Because, Mike, one would "never" get planning permission.
It is possible, with considerable expense, noise surveys and other expert evidence, to get full planning approval for a low use private strip, with people discreetly flying straight-in approaches and departures.
But the moment you propose to have training, with the heavy circuit banging which that involves, you can forget it.
OS and CA are very old established airfields.
It is a bit of a problem, because hard runway airfields do need training traffic to make money. The only exception would be one owned by a group of above average funded pilots.