I hadn't seen the Flora Mitchell version before, very cute.
Flora Mitchell was drawn to all those crumbling parts of Dublin where remnants of the gabled tradition could still be found in the 1950s and '60s. In this case, as the caption says, the subject mater of her painting was long gone and she had to base her representation of the Swift birth-place on a 19th century print, which itself may have been partly conjectural. It's interesting how she rationalized the profile of the curvilinear gable, a feature which is less clearly represented in the original print.
Whether the house in question was the actual house in which Swift was born in 1667 is open to speculation. Swift had the advantage of being venerated in his own time so it is possible that his birthplace was widely known to his contemporaries and never subsequently forgotten. On the other hand, in high-lighting the birth place of a notable citizen there is a vererable urban tradition of fixing on the nearest presentable house and letting time and repetition do the rest.
Could the house depicted in the Hoey's Court print date to the 1660s?
It is just possible. Brick construction was well established in the city by the mid-17 century and was becoming the norm in the grand expansion of the Restoration period which was occurring at exactly the time of Swift's birth and if the various claims we've made in recent times for the Clancarty House on College Green are true, then sophisticated curvilinear gabled houses were being built in Dublin in the mid-1660s, but whether this sophistication would have infiltrated the street-architecture scene in back-land locations like Hoey's Court already by the mid 1660s is another matter.
Even allowing for the uncertainty about the original gable profile, and the roof structure behind it, in most respects the elevation of the Hoey's Court house looks more likely to date to 1700 than 1660. If we're looking for parallels, the house that the Hoey's Court house most closely resembles, in the density of its façade fenestration and general detail and proportion, is the 'Ireton' house in Limerick [albeit a storey taller] which was a circa 1700 rebuilding and re-fronting of an older house and there is the suspicion of a re-fronting too about the Hoey's Court house, with the odd stepping of the first floor string course as though a pre-existing step in the floor levels inside had to be accommodated. Also the unusual profile of the gable might conceivably have derived from the need to screen some untidy existing roof profiles belonging to an earlier house. the 19th century print on which the Flora Mitchell painting is based compared with the façade of the 'Ireton' house on Nicholas Street in Limerick
Certainly, the heavy sash windows and the relieving arches over the windows in the attic storey would seem to link the façade to the main phase of the Dutch Billy tradition, which still leaves open the possibility the house behind this new façade may well have been mid 17th century and conceivably therefore the house that himself might have been born in.
In any other city, this would all have been researched and resolved and there'd be access to the excavated basement from the delightful little museum and coffee shop now sitting on the site.