I'm not saying it's a very common method (maybe, maybe not), but cycle timing is often included in "birth control" stats and used as a secondary measure with other forms of birth control.
You can just look up these numbers, of various forms of birth control. That exclude cycle timing.
And none of them say it’s “most people”. Perhaps some unmeasured population uses it as a secondary method, that’s total speculation and kind of beside the point
Are we (or the grandparent) talking most people at one point in time, or most people at any time during their life? If it's the latter, then I do think it's a valid claim, but I haven't seen any stats that cover those sorts of numbers. Eg have more than 50% of people in a committed relationship never used the timing method? Most of the couples I am good friends with have used the cycle timing method when they want another kid, want to delay it, but are open to having one sooner. This is also commonly combined with secondary measures, such as condoms, pulling out, or even abstinence during the fertile window (why measure cycle timing if you're not changing your behavior, using another method, based on the observation).