While there was some pre-production, I disagree that vaccine supply is the only binding constraint we have going on. And the lack of sane regulatory approvals and sane distribution setups has been heartrending.
If you look at https://www.bloomberg.com/graphics/covid-vaccine-tracker-glo... and scroll down to "U.S. Vaccination Campaign" you will see that in the US 74.5% of the vaccine doses that have been delivered have actually been used.
This could happen for a combination of two reasons, as far as I can see: either delivery is ramping up so fast that most of the delivered doses were delivered very recently, or because people are doing a terrible job of using doses that get delivered.
My impression is that the latter is a larger factor than the former, so far, based on the wide variation by state in the "Supply used" column. West Virginia has used ~96% of their supply. D.C. and Rhode Island have used only 60% (and it's not like their populations are larger or more spread out than West Virginia's!). And I should note that I can imagine various poverty-and-social-whatever issues D.C. might be facing, but they don't obviously apply to Rhode Island in the same way.
Things like http://www.arnoldkling.com/blog/virus-update-3/ item 3 are pretty common: crazy vaccine-distribution setups meaning people who want vaccines can't get signed up to get them, while vaccines are getting thrown out due to spoilage.
Things like https://www.nytimes.com/2021/02/10/us/houston-doctor-fired-c... are, I hope, not common, but you can bet that case will have a chilling effect on efforts by doctors to avoid wasting vaccines.
https://www.theguardian.com/global-development/2021/feb/14/w... includes this incredible from a major vaccine manufacturer: "Instead we have a patchwork of approvals and I have 70m doses that I can’t ship because they have been purchased but not approved. They have a shelf life of six months; these expire in April."
Does that sound like a manufacturing supply constraint to you?
Similar is https://marginalrevolution.com/marginalrevolution/2021/01/ap... where we have a vaccine, and a factory that can produce it, and the vaccine is approved in the UK, but it's not approved in the US. This is also not a supply constraint.