'When did foursomes become posh' is one of the more intriguing sub-fields in British golf history. As has been noted, many clubs beloved of GCA including the Sacred Nine, Huntercombe, Rye, Aldeburgh, Brancaster and Woking are pretty much two-ball only, certainly for member play. The Senior Golfers' Society (disclaimer- I belong) has an amazing fixture list of over 100 matches each year, almost all of which are foursomes.
As Sean says, foursomes are a cracking format for a 36-hole day, not least in that all participants can hit all the teeshots. But foursomes do presuppose that everybody plays a reasonable amount of golf, and probably to a reasonable standard. The British golf calendar used to start each spring with a welter of foursomes events, including the Sunningdale Foursomes (scratch-ish) the Halford Hewitt (huge team event for 640 old boys of British public (i.e. private) schools, and the Winchilsea Foursomes (handicap) at Harlech. These all still exist, but are no longer reported in the national press as they routinely were in the last century.