Jack's use of Bunburyism is no different than Algernon's, although Jack criticizes it.
Jack, in his own words is "Earnest in the country, and Jack in the city". His situation is that he is a ward to a young lady, the manager of a huge estate, and in general he is a man with responsibilities in the country.
He, however, needs to lose himself and finds freedom in the city. In London, he lives at Albany B4, changes his name to Earnest, and does all the wicked things that he cannot do in the country: Runs huge restaurant bills at restaurants, has creditors haunting him, and just enjoys the pleasures of dandyism.
It is as Earnest that he meets Algernon- the original bunburyist. In Algernon's case, he invents an invalid friend called Bunbury, and claims to go to take care of him when, in fact, it is just an excuse to go party around town. This is what validates the argument that Jack's use of bunburyism (disguised as an evil twin brother in the city named Earnest) is no different than that of Algernon's although Earnest thinks that Algernon, by doing this, is immoral.
As far as implications of Jack's personal life and his family, what this might mean is that Jack was simply bored of playing a paternal role to Cecily and even more overwhelmed to have been left with the many responsibilities of money, land, and estate. He was not an old man, and yet he had to find his fun only by escaping his life of seriousness and get in all sorts of scrapes in the city. This was maybe his only way to keep his youth from fading, and keeping his inner heat going.