Not necessarily that, but it becomes a liability.
If he doesn't develop it or support it, then the lockdowns last forever.
If he supports it but doesn't develop it, the lockdowns last forever (or at least until his term ends)
If he doesn't support the vax, but develops it quickly then why develop it at all. All the left becomes suspicious, then those on the right also become a little bit suspicious. Nobody takes the vax, Lockdowns still last forever.
If he supports the vax. Suspicions on both the left and the right are greatly alleviated. The vax appears successful (on paper) and the lockdowns end. He needs to continue to appear to support it until the risk of lockdowns returning are gone. This was the only path out.