>>The thought of "middlebrow dismissals" came to mind when I saw the ensuing discussion. Now I don't think I fully understand what is meant by the term

It's a spin-off from the term "highbrow", which means "intellectually stimulating". http://en.wiktionary.org/wiki/highbrow

A middlebrow dismissal means you are giving it only a half-thought before dismissing it, as opposed to critically thinking about it from multiple angles and offering an opinion that can result in a highbrow conversation.