I honestly don't think that everyone cares about the size of the company they're working for. I'm currently applying for internships for this summer and I am far more interested in what kind of work I could be doing than how big the company is. I don't think that's uncommon.
I know that many people don't care about the size of the company they're working for, but turn it around: Do you think that employers care about whether an applicant cares?
To make this perfectly clear, let me propose this interview question: What is more important to you: The company and culture or the work that you'll be doing?
I think that people being people, they will prefer applicants who honestly are more interested in the company and culture than the work. This is absolutely true of BigCo, and applies to startups as well. What if the company needs to pivot and you are no longer working on whatever excited you about coming to work on day zero? Will you start looking for somewhere else?
Well, I'm a student, so it is entirely possible that I underestimate the importance of company and culture. That being said, I'm not sure why I should care more about culture than my actual work. I obviously want a certain work climate (no annoying micro-management, smart and friendly co-workers etc.), but that should be possible in big companies in startups, no?
> What if the company needs to pivot and you are no longer working on whatever excited you about coming to work on day zero? Will you start looking for somewhere else?
Honestly: If it is a massive change for the worse, why wouldn't I? Example: I am interested in algorithms and low-level bit-fiddly stuff. I don't particularly care about HCI and web design. If I was to join a company that works on, say, databases and file systems, I would probably do it with the expectation to work on the kind of problems I find interesting. Would it really be unreasonable to look for something new if it turned out I could no longer do that?
It isn't for me to say what you should look for. I can tell you my experience of what employers want, but then again most employers probably want .NET experience, so what most employers want isn't the most important thing. The question is, how do you find the one employer that's right for you!
One way to find a middle ground between your interests and a company's concerns is to restate your interest in terms of the company culture. For example: I'm looking for a team that owns its stack and uses that ownership to do interesting and disruptive things with its product. For that reason, I'm less interested in the size of the company than in the way it approaches technology. If a team is working from algorithms and bit -twiddling on up, I'm sure I'll be happy.
That answer might apply to any company from Apple on down to a YCombinator startup.