Nearly but not quite, it wasn't almost everyone: those who booked via travel agents - which was a big chunk of the trade then - would get their travel agents to reserve seats for them. And that was one of their selling points which still limps on today, some old fashioned TAs can get seating for free on restricted tickets which is an avenue for those above who don't like the policy. Those who booked directly with BA could either do so in the former BA travel shops, or as you say over the telephone. However not many people bothered in those days, and in any case if more than about half the aircraft had their seats reserved the boards would come down and only TAs could make further seating changes. Many BA/BEA flights did not allow seat reservations full stop.