You forget the last week of the previous administration when they told the states a big shipment was coming, but there was no big shipment to ship.If you look at the chart of vaccination you can see it steadily increasing from December till now. The efficiency of last mile distribution/delivery increased throughout that time, nothing magical happened in that regards on Jan 20th. The limiting factor quickly became what everyone expected to be the limiting factor: How much vaccine is manufactured. Most distribution kinks were detected and state and local authorities adapted distribution accordingly.
Or when they panicked when they heard about states having a backlog of doses that were not being given out and so they breathlessly told the states 'jump to everyone 65+!!' when that wasn't the issue at all. The issue was people at all levels holding back 2nd doses (due to lack of communication of a plan... because there wasn't a plan) and states putting aside and being real slow about vaccinating LTC facilities. But the Feds didn't know that because they weren't keeping track, like they should have.
States that followed the 65+ recommendation wound up putting all their 1b and 1c people who weren't over 65 at the end of the line and created a panic because then the states were overwhelmed with people looking for doses that weren't there.
Don't forget that there shouldn't have been so few doses in December because the previous administration was promising 20 million doses in December and failed miserably to meet that goal.