The abiding memory of much of the reading I've done is that requirement for rough-field performance, and how much of an expense/compromise it caused.
Perhaps understandable, to a degree, given the nuclear era and the expected targeting of air bases by the Soviets. Nevertheless, subsequent aircraft (MRCA, for one) didn't go after the same requirement.
The mention of a hovercraft trailer brings to mind something else: was there not a VTOL platform also associated with the TSR2 programme?