For the same reason he didn't stick with her (or Kaido) as a pirate after the God Valley Incident: he despised his former crewmates and their ideals/(i)morality (Xebec was likely the "glue" that held all those crazies together). Whitebeard's concept of "family" isn't related to blood or anything like that, he achieved his dream the moment he formed his own crew. He was more aligned with the likes of Roger or Shanks, but even then they had their feuds.
As for Miss Buckin, she probably mixed WB's genes with hers, with the help of Vegapunk, to create Weevil. It's very unlikely that he had a son with her by "traditional means", Marco even said that WB never talked about anything like that (however, he knew that Buckin was a former crewmate of WB, so he apparently revealed at least the most important things to his subordinates).