Yes, correct.Most recently Dr. Fauci predicted that the vaccine will be available for everyone by mid-april I believe. If that is true, the biggest factors are how many vaccines can be administered per day and how many people actually get vaccinated.
I had read somewhere that only around 50% of adults have said they actually want to get a vaccine. So that's 125M people in the country (250M total adults). And at the rate we're going and want to ramp up to, the idea was that by March they'd already be running out of that 125M people to give vaccines to.
So maybe Fauci's comments are taking into account the slow ramp-up of vaccine manufacturing. Not sure, but I think supply is lagging at the moment. I saw where a lot of these "mega sites", at least in California and New York, aren't operating anywhere near full capacity.