In the UK here. You're pretty much right; the goal as stated was to get a first dose into all of the clinically vulnerable people (long term illness, diabetic, severely asthmatic, etc) and those over 50 before moving on to second doses. IIRC
those groups accounted for about 99% of deaths from covid.
In other words, the supply was enough to make sure that all of those people could get one dose in twelve weeks, so that's the dose interval that was set.
That goal (quickly vaccinate all of the at risk population with one dose) was pretty much achieved. I guess it remains to be seen whether it was the right goal, but early indications (deaths/day down from 1,000 to 10 or so) are positive. (Obviously correlation is not causation and so on, and there undoubtedly a ton of confounders.)