Vivek Ramaswamy’s move this week toward a run for governor of Ohio did not clear or freeze the Republican field the way someone so closely and visibly aligned with President-elect Donald Trump ordinarily might.

The entrepreneur will continue leading Donald J. Trump’s government efficiency project before turning his attention to a campaign. He has encountered some turbulence with Mr. Trump, and with Elon Musk.

Donald Trump’s ally, Vivek Ramaswamy, was bypassed for a Senate appointment in his home state of Ohio despite encouragement by the president-elect, foreclosing a political possibility for the entrepreneur at the start of the new administration.