The Club was most widely known a few decades following the Restoration.
In the shadow of the birth of the first political parties (in the modern sense of the word), and with a nation under continual threat of continued violence from Jacobites loyal to the King (who was in exile in France), political pamphleteers and political panegyrisms came to write constantly of the value of setting aside violence and engaging with rational speech in the political discourse of the era.
It was in this context that publications about the Calves' Head Club came to take on an increasingly exaggerated and outrageous tenor, and their provenance was increasingly of the Grub Street variety, i.e., they were primarily written about by Ed Ward in sloppily expanded editions throughout the first years of the 18th century. The cabal of the Calves' Head Club, expressing support for the beheading of the King, represented an ethos that was antithetical to the political culture of England. It also made for pretty funny reading by an author who was popular as a low-brow humorist and whose tales of drunken wanderings through London pubs were widely read (and shared with non-literate friends by those who could read) in London.
The club allegedly continued to meet until 1734, when diners who were feasting on January 30th -- a day that should have been one of mourning or at least quietude in honor of the death of Charles I -- were mobbed because they was believed to have been the Calves' Head Club conducting their disgraceful rituals. Whether these people were members of the club, or whether the club had ever met after the Restoration itself -- if it ever had met -- could be reasonably called into question given the source material.
What is certain is that the outrage-provoking behavior of the Club, as described by Ward, reached wide popularity years after the Restoration, when anti-Charles I sentiments were long out of style. The Club members became a popular symbol of anachronistic anti-monarchists against which Londoners, fearful of Jacobites who preferred the return of the exiled King in France, could define themselves.