Look at the companies that you admire and that show the ability to innovate... chances are they are run by people with technical backgrounds.
Example: AMD is run by Lisa Su, whose background is in EE. HP was run by MBA holder, Carly Fiorina.
You're judging them on their compatibility with your personality, not on the actual knowledge that an MBA or other sort of business degree provides. It's really easy for engineers to completely dismiss the kind of knowledge (including emotional knowledge) that people in sales, business, and marketing possess. They aren't just some dummies who don't know how to code.
It also shouldn't be surprising that pure engineering companies like AMD are run by engineers, and while HP is an engineering company in some form, it's also more of a business services company. They do things like contract out desktop support services to large corporations or the government - that is much less about engineering and technical ability and more about providing continuing services, providing a business relationship.
It's actually not a lot different than hiring a custodial firm. Why wouldn't someone with a business background run HP? HP's commodity hardware business doesn't actually make a whole lot of money from what I remember.
Remember, many people who are candidates for an MBA already have a non-business degree, and they're looking to get a fundamental business education.
So a wonderful example of that might be Tim Cook, who has a bachelor's in Industrial Engineering as well as an MBA, and runs the most valuable tech company in the world.
Many people say he's not as good as Steve Jobs (due to the lack of personality cult), but then again, Apple is now making more than double the revenue that it did under Steve Jobs with a revenue mix that is much less dependent on the iPhone. The fastest growing segment is services, which is far more profitable and less risky than hardware (ask Pebble, BlackBerry, or Fitbit about that).
For example, in hard market conditions, or grim financial periods usually the CFO takes the leash as the CEO, in times where innovation is required a CTO takes the leash, and in times of abundance and room for growth CMOs come forward - I know it's not that linear, but in different stages companies bring forward different people with different backgrounds.
Also you can innovate in a lot of ways other than technical ways - brand, business model, operations, supply chains, distribution, market development, the list goes on.
>Example: AMD is run by Lisa Su, whose background is in EE. HP was run by MBA holder, Carly Fiorina.
You're looking at the wrong thing. You're looking at the companies, their products, etc., and how much you admire those companies and what they produce.
What you're not looking at is executive compensation. You obviously don't think much of HP when Carly was running it, but how much money did she make in that position? That's what's really important, not how the company did, what products it made, or what you personally think of any of that. For Carly, the only thing that matters is how much money she got, so that's the only metric that matters when discussing whether her MBA has any value. Did Carly make more money than Lisa Su? If so, then Carly's background was more valuable to her.
I work as a Software Engineer.
If you were in a role that required some basic knowledge of the business, and you didn't previously have that knowledge, I'm sure it would come in handy (i.e. basic accounting, finance, and legal concepts).