How did "bunburying" end up playing a major role in The Importance of Being Earnest by Oscar Wilde?

Bunburying is a word Algernon coins to represent the act of using a fictitious person as an excuse to avoid unpleasant interactions with others or to gain a sense of freedom for a short time.

Algernon invented his dear friend "Bunbury," a man prone to sudden, unexplained illnesses, as a means of avoiding social engagements with his Aunt Augusta and her family. He soon discovers his friend Jack has likewise used a nonexistent brother named Ernest to gain freedom from his duties in the country. As Jack is spending an evening under the alias of Ernest in the city, Algernon stumbles upon his real identity and proclaims, "I have always suspected you of being a confirmed and secret Bunburyist; and I am quite sure of it now." He uses this information to bunbury in the country to meet Cecily.

The problem arises because both women the men desire to marry believe they are engaged to Ernest. Gwendolyn believes Jack's name is actually Ernest. Cecily, Jack's ward, is made to believe Algernon's name is Ernest when he makes his bunburying trip to the country to meet her. Both men have to come clean about their deceit.

Without the concept of bunburying, Cecily would have never met Algernon. While Jack and Gwendolyn would have met, Jack likely would have never learned his true identity as Algernon's brother. Most importantly, he would have never learned "the vital Importance of Being Earnest."