Then, almost as quickly as it began, the controversy subsided. And several weeks later, Mr. Trump decided not to seek the Republican nomination. Though he continued to do well in polls, he seemed to be more focused on his reality television pursuits.

Now, Mr. Trump almost assiduously refuses to discuss the topic, which, according to several people close to him, was always more about political performance art than ideology.

“I don’t talk about that anymore,” Mr. Trump told the MSNBC host Chris Matthews after a Republican debate last year.

Raising questions about the president’s birth certificate — and even threatening to send a team of investigators to Hawaii — had served its purpose, raising Mr. Trump’s political profile and, whether he knew it or not at the time, providing him with the rudimentary foundation upon which he built his 2016 campaign.

He even skirted close to birther innuendo after the massacre in Orlando, Fla., last month, calling into “Fox & Friends” to insinuate that Mr. Obama might sympathize with Islamic extremists. “He doesn’t get it or he gets it better than anybody understands,” Mr. Trump said.

But for all of his fascination with the president’s birth certificate, Mr. Trump apparently never dispatched investigators or made much of an effort to find the documents.

Dr. Alvin Onaka, the Hawaii state registrar who handled queries about Mr. Obama, said recently through a spokeswoman that he had no evidence or recollection of Mr. Trump or any of his representatives ever requesting the records from the Hawaii State Department of Health.