I thought the RAF could guarantee air cover anywhere in the World and there would be no need for the Royal Navy to have its own air power.
That was the proposal up to 1965. However, contrary to popular belief it was the RAF "Island Hopping" strategy that was rejected and the Navy carrier/amphibious strategy which was accepted - hence the cancellation of the TSR2 and all RAF strategic reach.
It was the subsequent Defence White Paper of 1996 which reduced the UK to a strategy of having no force projection capability East of Suez and the cancellation of CVA01.
RUSI:
Inter-service rivalry: British defence policy, 1956-1968
Meanwhile, as the RN and RAF squabbled over the expeditionary role, Army interest east of Suez focused on the requirement to fight insurgencies in Borneo and Aden and the need to support them in this often tied down expeditionary forces, particularly the navy's amphibious ships and helicopters, reducing their availability elsewhere.[18]
The JSSF concept was well-suited to British defence needs as they appeared in the early 1960s and on this basis the Macmillan government rejected the RAF island strategy and agreed to build a new large aircraft carrier, CVA-01. As is well known, the carrier did not progress beyond the drawing board still less did the RN get the second ship they had anticipated. In 1966 the Labour government, bequeathed unsustainable spending plans by the previous administration, cut the programme.[19] This did not represent a victory for the RAF's alternative vision as much as an overall reduction in British aspirations. The island strategy did not, could not, provide the flexible range of options offered by the JSSF but the government decided that it did not require such options and with ambitions suitably reduced, the RAF plan would suffice. Quite how a total of twelve F-111 aircraft would truly have served British interests, out of sight and out of mind at airfields remote from many potential trouble spots, was never put to the test.[20] Within just two more years it was decided to withdraw from east of Suez altogether. There were insufficient funds even for this token capability.[21]