The problem was even if his plays were popular as they were in his lifetime and sometime after, once the Puritans seized political power, they were effectively suppressed and thus…went underground.
And popularity doesn’t always correspond with being regarded as respectable within the academic or official establishment of the period.
One example from Chinese literature is The Dream of Red Chamber/Story of the Stone which is currently regarded as such a notable literary classic in Chinese lit in China and abroad that there’s an entire subfield devoted to the study of that one novel.
However, despite it being extremely popular during the lifetime of the author and sometime afterwards, back in the day(~18th century), it was regarded by the prevailing establishment as a “lowbrow guilty pleasure” similar to reading a “Penny Dreadful” or a Dime novel in the 19th century Britain/US…or in a more modern context…reading the National Enquirer in the print newsmedia context. Especially if the one doing the reading is a member of the educated establishment as many of the readers of Red Chamber were back in the day.