There was a CEO survey that came out just as I was graduating from college. They asked what was the most important factor in their career. The #1 answer, North of 60% of responses, was "luck." Like you, they survey went on to say that it wasn't that they felt unqualified, but just that there were other qualified people they knew, who didn't get the same opportunities. So yes, your observation is right. The other people know it, too, and it works at every level of an organization.
So while it is always good to take an honest look at yourself, you do need to understand there is no firm path to results; it might be useful or necessary to ante into the game, but there is still a lot of luck. I say this, because despite your first observation, you seem to be searching for a silver bullet for your situation, when there is none.
Looking at the details, you mention in several ways that you aren't comfortable networking, while this may be part of an MBA,, there are far simpler ways to go about it. I have no idea how they are operating in a Covid environment, but I would recommend looking into Toastmasters or Dale Carnegie.
Which isn't to say an MBA won't help. How comfortable would you be hiring and firing people? Facing a challenging 10% department budget cut? Deciding on an advertising campaign or changing an account offering? While your first management job won't involve all of that, that is the path you are determined to head down. While a background in finance is an excellent starting point, particularly for a bank, an MBA can also expose uou to these other disciplines, so you won't have your first encounter with them be a point of decision or crisis in your job--and hopefully, it will help you make fewer mistakes when the time comes. But management itself is a discipline, and there are many people who wanted the bigger paychecks of management, without realizing that they were saying goodbye to the disciplinw they loved, one promotion at a time.
As far as school reputation is concerned, I don't see a problem with a local school with a good reputation. If you had said you were looking to leave your current org ASAP, and that it was way too small for your ambitions, then shooting for the best alumni association you could might be worth it. But your requirements seem to be directly about the training you want to get, not about the contacts.
For what it's worth, I got my MBA at a regional campus of a Big 10 school. I also considered a program at the main campus of another Big 10 school. A lot of hard-charging people in my megacorp employer were going there. But it was a 2-week on, 6-week off quasi-executive MBA, which due to the time structure meant these people still kept their day jobs. I have no idea how they did it, but let's say I received plenty of work emails from them past midnight, and it still took 2 years. My program, though less well-networked, was still well regarded. And the smaller campus attracted a lot of professors who were former practitioners in their field. So a lot of their examples, either from personal experience or pulled from media sources, were likewise very practical. I loved it--it was just what I was looking for. And my classmates were a diverse bunch too; yes, there were a number of us from the several megacorp in the area. But there were also a number of entrepreneurs, and people working at nonprofits. I loved that, too, although I saw no direct use in it for career development.
Good luck in your choices. The first step is to be self-aware, and knowing your career is in your hands. Many people at larger organizations never even get to that point. So, you have already taken the first step.