Harlan Band had turned a corner. He was 29 years old and back in school, catching up on algebra and Shakespeare, earning A’s, seeking to put a lost decade behind him and looking ahead to studying medicine.

That was the story Mr. Band told his family. He also said it was the reason he sought help from Done Global Inc., an online mental-health company that prescribes stimulants like Adderall in video calls as short as 10 minutes.