Before giving Trump too much credit, remember he was out of office before the vax was in widespread distribution. Based on his performance on this issue before he left office, I think he would have mandated the shots if Birx and Fauci recommended it.
That's an assumption you can't prove. It's simply speculation. You fail to recall that Mike Pence was in charge of the task force. We still have no knowledge as to what recommendations he made to Trump, nor those recommendations that Trump rejected. Pence was never asked about any of it, and likely never will be questioned about it. Democrats had no interest in investigating any of it after they took control of Congress in January 2021.
Besides Birx and Fauci, other members of the Covid task force were Vice Admiral Jerome Adams, Surgeon General; Alex Azar, Secretary of Health and Human Services; Stephen Biegun, Deputy Secretary of State; Robert Blair, Senior Advisor to the White House Chief of Staff; Ben Carson, Secretary of Housing and Urban Development; Francis Collins, Director of the National Institute of Health; Ken Cuccinelli, Acting Deputy Secretary of Homeland Security; Kelvin Droegemeier, Director of the Office of Science and Technology Policy; Thomas J. Engels, Director of the National Institute of Allergy and Infectious Diseases; Admiral Brett Giroir, Assistant Secretary for Health; Joe Grogan, Director of the Domestic Policy Council; Stephen Hahn, Commissioner of Food and Drugs; Derek Kan, Executive Associate Director of the Office of Management and Budget; Larry Kudlow, Director of the National Economic Council; Chris Liddell, White House Deputy Chief of Staff for Policy Coordination; Peter Marks, Director of the Center for Biologics Evaluation and Research; Steven Mnuchin, Secretary of the Treasury; Robert C. O'Brien, National Security Advisor; Sonny Purdue, Secretary of Agriculture; Matthew Pottinger, Deputy National Security Advisor; Robert R. Redfield, Director of the CDC; Eugene Scalia, Secretary of Labor; Joel Szabat, Acting Under Secretary of Transportation for Policy; Seema Verma, Administrator of the Centers for Medicare and Medicaid Services; Robert Wilkie, Secretary of Veterans Affairs.
On February 26, 2020, Pence replaced Azar as the Chair of the Task Force.
Trump announced the formation of Operation Warp Speed on May 15, 2020. The first person received the shot on December 14, 2020, at which time, the election had already been stolen from Trump. The whole shebang was handed over to Biden on January 20, 2021, and the rest is history.