Because Mr. Castro never ran for statewide office, he has not built up a big email list of supporters, and he still has not met the 65,000-donor threshold to qualify for the first Democratic primary debates. In the first three months of the year, he raised only $1.1 million (though his campaign said it had raised over half a million more by mid-April).

[Here’s a look at the early fund-raising numbers for the whole 2020 field.]

“I don’t believe that from the presidential perspective, it was a mistake,” Gilberto Hinojosa, the chairman of the Texas Democratic Party, said about Mr. Castro’s decision not to run statewide. “But I do believe that, like Beto, had he run for one of those offices, that he would have done really well even if he wouldn’t have won.”

Mr. Castro is more circumspect, betting that voters will come to appreciate his background and executive experience, eventually. For now, he said, he is focused on building out his campaign team and increasing his grass-roots support. He has pledged to visit all 50 states during his campaign.

“People are going to have their moments,” he said, without naming names but perhaps alluding to Pete Buttigieg, another 2020 candidate and the mayor of South Bend, Ind., whose popularity has grown substantially in the past month. “I would rather have my moment closer to the actual election than right now.”

He ticked off former Republican candidates who surged early, then flamed out, naming Herman Cain and Michele Bachmann.

“I’m sure that’ll come,” he said, “but I don’t just want two weeks in the sun.”

Mr. Castro was born and raised in San Antonio. His grandmother emigrated from Mexico when she was seven. His mother, Rosie Castro, was a prominent civil rights activist in the 1970s.