It... would not. We need half a million (edit: billion of course, oops) doses to "end this thing", and those just aren't going to be there from any combination of manufacturers. If that kind of result was really on the table, a new administration and congressional leadership would be moving mountains to get that win.
But yes: AZ should probably be approved faster than it is and I don't know what the delay is.
They are waiting for the results from a larger US-based trial.
I'm not sure why they aren't going with the same trial data they used to submit requests elsewhere, but it could have something to do with the FDA approval process. The FDA evaluates drug approval requests the way a lot of people were saying the FAA should have evaluated Boeing's 737 Max certification application.
The FDA takes your trial data and then does their own analysis of safety and efficacy. They also have an independent advisory committee that looks over both the company's work and the FDA's work. Many other places, such as the UK, rely more on trusting the company's analysis.
The AZ vaccine was the one that had the error in running the trial that gave some people half of the dose they were suppose to get, and the results suggested that results were better for them than for those who got the full dose. The trial wasn't designed to test half dose vs full dose, and so the set of people who ended up getting half vs full are not really a good set to draw conclusions from.
Perhaps AZ thinks that submitting based on a later trial without that error will overall lead to faster approval than submitting an earlier trial that would probably take longer for the FDA and the independent advisory committee to analyze?
The results weren’t even negative. A subset of their trial group received a different amount of vaccine which seemed to show greater efficacy later.
The issue, and thus the “mistrust”, is that if they messed up that rather critical part of their clinical trials then what else could they have messed up?