Actually, I'm not sure I consider the British Open as a major from the early 1930's until around 1960. I think most of the top Americans did not play there much. e.g. from 1937 to 1960 Sam Snead played it once. (And won.) Hogan played it exactly once in his entire career. (And won.) Nelson played it twice, in 1937 and 1955.
I think it was considered a big deal when Hogan won the Masters, US Open and British Open in 1953. He got a ticker tape parade down Broadway for the exploit. I am not sure whether the British Open had equal status with the US majors - it probably didn't - but I think it was a bigger deal than say, the Western Open, which used to be considered a major.
After winning the Masters and US Open in 1960, Arnie said that if he won the British Open and the PGA it would be the equivalent of what Bob Jones did in 1930, thereby defining the modern grand slam. Arnie isn't that imaginative; the British Open must have had a special status even though top US golfers ignored it. I don't think anyone contradicted his claim that the British Open belonged in the grand slam. Of course Arnie's and then Jack's participation is what solidified the significance of the event with the public.