For sure pubs ran through prohibition.
McSorley's Old Ale House on the Lower East Side of Manhattan was definitely fully operational during prohibition. You can read about it on Wikipedia, but my preferred reference is Joseph Mitchel's "McSorley's Wonderful Saloon"
Such bars exist but I think these bars are less likely than average to be those bars. Regrounding after prohibition was particularly likely for these locations and many of them are going to be smaller than average.