Multiple reasons:
Big companies are frequently composed of components with distinct culture, benefits, legal entity, and so on. People get to know one part, then assume that all parts are that way. I'm in a component with compensation that is better than average for the big company. The association isn't beneficial.
Using the big company name means adhering to corporate branding requirements. People could somehow imagine me to be issuing official corporate communications, which is far from the truth.
I might get more responses, but would I want them? We're fighting to keep out the toxic people who want to focus on politics. We have important work to do.
There is an absurd Glassdoor review out there. If that is to be believed, we pay highly experienced cleared specialists about as much as they'd make cooking food at In-N-Out or Chick-Fil-A. Our office would be empty if that were the pay being offered.