They want that kind of status, and envision a chance at making a lot of money.
They rationalize it afterwards because that doesn't make a good story and why not?
But these explanations only aggrandize the startup creator and offers the wrong way to think about how to start one's own company.
However, I would question the idea that the only way to think is in terms of "problems." There are many valuable things people can do beyond solving problems. Solving problems takes care of our basic needs, and there's definitely something to be said about the relative importance of solving problems vs. doing other things (creating art, for example), especially if you want to "make the world a better place" (tm), but I feel it's still a bit limiting.
That's why I like the Y Combinator motto "Make something people want." Start from the want, not the "problem." Solving problems is just one of many things people want.
Are there many examples of startups for the sake of startups that succeeded?
To be fair, I do feel like there is a certain population disingenuous founders who weave a backstory around the genesis of their startup...but the population, by my estimation is small.
I've known a lot more that start companies in things they're passionate about (e.g. video games) and struggle.
The feel good mantra of our time is damaging. It certainly was to me.
But I think that is the main point, seeing something that is an oppurtunity.
I was more reffering to people who "play startup" for startups sake, because it is cool or the done thing.
If there is oppurtunity and solution you don't need to be passionate generally.
No one started a sewer cleaning service because they love cleaning sewers.
My view is you should always start with a problem that is significant to you, and solve it. I know that making a living is important, but if the problem applies to more than just you and you can charge actual money for it, I feel it is worth trying. The next question is if you can afford to take on such a risk, as it is a great one. This is substantially more risk than thinking of something to think of something because you want to run a company and/or make money. In the end, if you solve your own problem, you at least have that if you lose completely or go nowhere with it.
This of course is a rosy view and things like debt and moral responsibility to employees increase the cost of such a strategy. From an optimistic point of view, this is how I would hope people try to start their companies - with good, utilitarian intentions.
I for one am so tired of all the companies that are in search of a problem that does not need solving. I am also tired of all the cash given to companies that solve some problem, but don't meet the 2nd part of what I said - make money. If you can't tell me how your company will make money in 1 sentence, you have at best a hobby, not a real company. Unfortunately, we have a billion dollar industry of glorified hobbies.
I also feel the same about advertising-based funding, but that's another minefield and to some degree one could argue at least is necessary in some spaces - ex: TV, newspapers. Too many people use ads as a hand-waiving excuse to explain how they will make money.
Unfortunately, the cold-hard capitalistic view of things and use whatever advantages you can get. Sometimes this means screw the morals, passion, soul, ethics, etc. But you know it does work and some people convince themselves they are happy. Maybe they are, but for many of us like me, I seek more intellectual purity than financial glory (as much as I want the latter).